Godflesh

Godflesh are an English industrial metal band from Birmingham. The group formed in 1982 under the original title O.P.D. (later Fall of Because) but did not release any complete music until 1988 when Justin Broadrick (guitar, vocals and programming) and B. C. Green (bass) renamed the band and decided to use a drum machine for percussion. Melding heavy metal with industrial music and later with electronic music and dub, Godflesh's sound is widely regarded as a foundational influence on other industrial metal and post-metal acts and as significant to both experimental and extreme metal.

The band signed to Earache Records in the late 1980s and released their debut album Streetcleaner (1989) to contemporary and lasting acclaim. After the release of their sophomore album Pure (1992) and their major label debut Selfless (1994), they started experimenting with live drums as well as hip hop and breakbeat sounds. The resulting albums, Songs of Love and Hate (1996) and Us and Them (1999), were followed by Hymns (2001), which saw a simplification of the band's style. Shortly after Green's departure in 2002, Broadrick ended Godflesh and pursued various other projects, such as Jesu. Broadrick and Green reformed Godflesh in 2009, releasing A World Lit Only by Fire (2014) and Post Self (2017) to critical acclaim. Their ninth album, Purge, was released in June 2023.

As pioneers of industrial metal, Godflesh's early sound was defined by harsh machine beats, production-emphasised bass, distorted guitar and sparse vocals delivered in a low, guttural fashion. The band performed for most of their career as a duo with Broadrick and Green playing over preprogrammed percussion, normally to a backdrop of apocalyptic scenery and Christian iconography. Godflesh's music has been regarded as especially heavy and grim, with Streetcleaner being named by several publications as one of the heaviest and greatest metal albums.

## History

### Formation and early years (1982–1988)

The band that would eventually become Godflesh, O.P.D. (Officially Pronounced Dead), formed in 1982 when B. C. Green and Paul Neville, two young musicians living in cheap council estate housing in east Birmingham, started experimenting musically alongside a drum machine. Soon in 1983 the band renamed to Fall of Because, named after a Killing Joke song and a chapter from an Aleister Crowley book. Later the band found a live drummer Justin Broadrick, who lived in the same council housing as Green and Neville, and joined the group in 1984 after organising a concert at The Mermaid in Birmingham. At that show, Fall of Because, Final (Broadrick's first musical act) and an early incarnation of Napalm Death performed before a crowd of twenty-five people. In the months following that concert, Broadrick joined Napalm Death as a guitarist and Fall of Because as a drummer and altered the latter's sound by introducing albums from Swans, Sonic Youth and Discharge to Green and Neville. Only fifteen at the time, Broadrick said he "usurped" their band.

Fall of Because recorded a demo titled Extirpate in 1986, which contained several tracks that would become Godflesh songs. Due to these recordings not being widely available until 1999, they were retrospectively recognised as "eerily" ahead of their time by Exclaim!. Later in 1986, Broadrick was invited to play drums for Head of David, leading to his departure from Napalm Death and soon after from Fall of Because in 1987. Then, in March 1988, he left Head of David for being, according to Broadrick, "too noisy of a drummer", and contacted Green in April to reform Fall of Because as a duo. In that reformation, Broadrick took over on guitar, and the band went back to employing a drum machine for percussion. It was then that the group was renamed Godflesh. Broadrick explained the new name by saying, "I heard someone once say that music is the voice of God. The word 'God' conjures something immense and inconceivable. The 'flesh' part is what effects you on a physical level. Our music is loud and destructive."

### Self-titled EP, Streetcleaner and Pure (1988–1993)

Inspired by the bleak urban landscape of Birmingham and the extreme music Broadrick introduced to Green, Godflesh took on a distinctly heavier tone than the primarily Cure-influenced Fall of Because. In 1988, the band established a presence in underground music by releasing their self-titled extended play (EP) through the Swordfish label. That EP, considered the source of industrial metal alongside Ministry's 1988 studio album The Land of Rape and Honey, combined programmed industrial beats, distorted vocals, low guitar and driving bass riffs to create the sound that Godflesh would become known for.

Shortly after the Godflesh release, the band recorded another EP titled Tiny Tears comprising four short, rough songs. Before Swordfish could release this EP, though, Godflesh were acquired by Earache Records, and the label's founder, Digby Pearson, convinced Broadrick and Green to shelve Tiny Tears and use the songs as bonus tracks on their first full-length album. The band agreed, and in 1989 they released Streetcleaner, which went on to receive critical acclaim and recognition as a landmark album in heavy metal music. Streetcleaner saw the reintroduction of Neville into the band, this time as the second guitarist, and it marked the band's first release of many on Earache. The album further defined Godflesh's sound, standing out from other metal releases with unusual production that emphasised mechanical beats and percussive bass over guitar. Streetcleaner is regarded as particularly heavy and bleak.

From February to March 1991, Godflesh were again in the studio, recording the Slavestate EP, which saw the band experimenting with dance and more electronic elements. Later that year in April, Godflesh embarked on their first tour of North America (a leg of Earache's Grindcrusher tour) with labelmates Nocturnus and Napalm Death. According to Mike Browning of Nocturnus, most of the shows of the 45-day tour were attended by 200 to 300 people. Godflesh missed the first concert due to permit issues, but they made it to the second date at the L'Amour in Brooklyn. The venue was full, and when Godflesh took the stage, their drum machine failed and the band could not continue. When a replacement machine was eventually found, Broadrick and Green hastily programmed in four songs to be ready for the tour's third show at the Channel in Boston. Despite these initial difficulties, the rest of the tour went well and Godflesh drew surprisingly positive reception. Slavestate was released in July after Grindcrusher had concluded. In August and October of 1991, both Broadrick and Green guested on Buried Secrets (1992), an EP by Painkiller.

With the successes of Streetcleaner, Slavestate, a concert opening for Nirvana and the Grindcrusher tour, Godflesh started on their second album, this time without Neville, who chose to focus on his other project, Cable Regime, which featured Broadrick as a recurring producer. To fill the void, Robert Hampson of Loop was brought in to play on half of the new album's tracks as well as on Cold World (1991), an EP recorded in the same sessions. The sophomore album, Pure, was released in 1992 through Earache and has since been recognised as an influential release in the post-metal genre. Musically, Pure was even more mechanical than Streetcleaner, further emphasising the drum machine and featuring production that augmented the percussion with a stark, bleak atmosphere. Though Godflesh's most overt experiments with hip hop and breakbeats occurred later in their career, Pure featured elements of both buried under the wailing guitar, shouted vocals and aggressively repetitive drumming. The album continued its experimentation with atypical song structures on its conclusion, "Pure II", a twenty-minute ambient drone piece with a submerged beat that, as AllMusic's Ned Raggett said, "hits like a distant cannon".

Despite Broadrick's dissatisfaction with the mixing on Pure being "not heavy enough", many critics regarded the album as unrelenting; in a positive review, Spin's Mike Gitter wrote that "Hiroshima was probably more fun [than Pure]". In support of the album, Godflesh planned to open for Ministry on another tour of North America, but instead ended up opening for the electro-industrial band Skinny Puppy on their Last Rights (1992) tour. Due again to issues with entering the United States, Godflesh were forced to cancel a number of these dates; they later returned to those markets as headliners to make the missed shows up. Broadrick retrospectively said that this era of the band stands as "the most honest representation of what Godflesh set out to achieve."

### Selfless, Songs of Love and Hate and Us and Them (1994–2000)

After a year of minimal activity in 1993, Godflesh were approached by several record labels. According to Broadrick, Danny Goldberg of Atlantic Records invited them to London and expressed his desire to acquire the band. About this period, Broadrick said, "They really thought that Godflesh could be the next Nine Inch Nails and that we would be selling out fucking stadiums. The buzz at the time was ridiculous. It outweighed the sales, obviously. It was all hype." Ultimately, the band ended up with major record label Columbia for the release of the EP Merciless in 1994, the title track of which was originally a Fall of Because song. Another EP, Messiah was recorded during these sessions, but it was not released on a wide scale until 2003 through Relapse Records. Later in 1994, the band released their third album, Selfless, which represented a shift in the group to a more high-end production approach and to a greater focus on traditional heavy metal riffs. Despite being the band's best-selling record with approximately 180,000 copies shipped, Selfless was deemed a commercial disappointment. This coupled with the ban of Godflesh's first major music video led to the end of their collaboration with Columbia.

Feeling abandoned after being abruptly dropped by Columbia, Godflesh were briefly directionless in 1995. In 1996, the band returned to Earache and created their fourth studio album, Songs of Love and Hate, which was Godflesh's first music made with a human drummer since the early Fall of Because days; Bryan Mantia of Praxis provided the aggressive, non-mechanical drumming. In retrospect, Broadrick believed Songs of Love and Hate marked the point where Godflesh lost sight of their original goal and started making "self-conscious" music. When it came time for the album's 1996 tour, Mantia made the move to join Primus, and Godflesh recruited Ted Parsons of Prong and Swans to perform on the tour in his place. Along with the album's follow-up remix release, Love and Hate in Dub (1997), Songs of Love and Hate moved away from Godflesh's industrial roots into experimentation with conventional verse-chorus format, hip hop, dub and drum and bass. The remix album was supported by a one-off concert on 4 October 1997 at The Garage, London where Broadrick operated a mixing desk, Steve Hough played guitar, Green played bass and Diarmuid Dalton provided support on a Moog synthesiser.

This experimentation continued and increased with Godflesh's next album, Us and Them (1999). While live drumming was dropped again in favour of percussive machines, Us and Them saw the group going further with electronics and drum-and-bass-oriented sound than ever before. Broadrick was quick to admit that he "hated" the album and that it was an "identity crisis". Retrospectively, though, he revised his thoughts, saying that his hatred was overstated despite him still having issues with the album. Shortly after releasing Us and Them in 1999, Godflesh began work on a proposed remix album, Us and Them in Dub. While this album was never released, two tracks from it appeared on the 2001 compilation In All Languages. Also in 1999, Life Is Easy, an album compiling Godflesh's recordings as Fall of Because, was released on the Alleysweeper label and distributed via Martin Atkins' Invisible Records label.

### Hymns and dissolution (2001–2002)

Following the release of the double album compilation In All Languages, Godflesh released their sixth studio album, Hymns (2001) through several different labels. The album again saw Godflesh with a human drummer; Parsons returned to provide the live percussion and, according to Broadrick, the inspiration for Godflesh to continue. Hymns marked a dramatic shift away from Godflesh's recent experimentation, moving instead into the realm of direct heavy metal, with only a few tracks revealing hip hop or electronic influence; Broadrick wanted Hymns to be more of a rock album than any other Godflesh releases. The album was recorded in a professional studio and a producer outside of the band was brought in to oversee the process, two firsts for Godflesh that Parsons believed to be mistakes. Despite receiving positive reviews, the final product of Hymns was ultimately dissatisfying for Broadrick, so much so that he restored it to a state resembling its demos on the 2013 reissue. With the band disheartened by Hymns''' troubled production and dreading an upcoming tour, the direct future of Godflesh was unclear.

In October 2001, the same month Hymns was released, Broadrick received a call from Green just two weeks before Godflesh were set to tour with Strapping Young Lad and Fear Factory. Over the phone, Green expressed frustration at having to open for young bands despite being in the business since 1982, as well as being forced into a tour for an album that was manipulated from the outside. Green tearfully left Godflesh to return to university and focus on his relationship with his partner. To cope with the departure of his friend, Broadrick devoted all of his effort to maintaining the band. It was quickly announced that Green would be replaced by former Killing Joke and Prong bassist Paul Raven. Though this configuration of Godflesh came up with some ideas for another studio album (namely the usage of a rhythm from Public Enemy's "Shut 'Em Down"), Broadrick was "always aware that a new album was never going to happen". Not long after this short-lived incarnation of Godflesh performed a handful of shows that Broadrick later said "felt completely wrong", he experienced a nervous breakdown the day before departing for another tour of North America, this time with High on Fire and Isis. He recalled the breakdown as a "real Brian Wilson moment" and said, "I felt paralyzed by the stress, which had been building for several months, and I literally couldn't get out of bed. I was numb and couldn't move, so when the car came to pick me up to take me to the airport, I ran and hid at another friend's house in Birmingham." All of the shows were called off, and Godflesh officially disbanded on 10 April 2002.

The cancelled tour caused a number of problems for Broadrick; his thirteen-year relationship with his girlfriend dissolved, and a bus driver who had been hired for the tour threatened his life. In a panic, Broadrick remortgaged his house and amassed approximately US\$35,000 to pay the driver and everyone else who had been affected by the cancellation. Broadrick, at a low point in his life, was left with little to do but work on new, different music. About the period, he said, "My only solace, my only escape at that time was recording the first Jesu album." "Jesu", the closing track of Hymns and, until 2014, the last original Godflesh song, ended with a hidden passage of unusual peacefulness and melodicism. Broadrick's next project, bearing the same title of Jesu, adopted that shoegaze sound and the new band's first EP, Heart Ache, was released in 2004. On promotional material for Heart Ache, a message from Broadrick read, "Godflesh is dead, long live Jesu."

### Reformation (2009–2013)

After Godflesh's dissolution, Broadrick and Green rarely spoke. Though there was no falling out between the two members, Broadrick assumed the project permanently dead, believing that Green's interest in it was gone. Regardless, Broadrick decided in 2009 to approach Green with an offer to reform, something that promoters had been pushing for years. Green responded to Broadrick's call within a few hours, saying he would love to do it. Both members quickly agreed that the most important thing for the reunion's integrity was that they return to a drum machine for percussion. In November 2009, the reunion was made public when Godflesh announced they would perform at the 2010 edition of the Hellfest Summer Open Air in Clisson, France. Asked in a February 2010 interview about Godflesh's future, Broadrick replied, "Godflesh will not commit to anything but Hellfest currently. I am unsure as to where we will go from there, if we go anywhere at all." He also revealed that the likelihood of new material was "fairly minimal", but that the possibility still existed; though later, in 2014, he insisted that "ultimately, the whole reformation thing was pretty much fundamentally about wanting to make new music". On 18 June 2010, Godflesh performed for their first time since 2001 at Hellfest. The show was fraught with technical difficulties, ending up as a forty-minute set instead of the intended sixty minutes. Despite the frustration of the return performance, Godflesh continued to play festivals throughout 2010 and 2011, including the 2011 edition of the Roadburn festival in Tilburg, Netherlands, where they performed their first album, Streetcleaner, in its entirety. This performance was later released as the group's first live album in 2013.

In December 2010, Broadrick revealed to Decibel magazine that Godflesh were gradually assembling new ideas for a studio album. He explained, "It's something we're discussing all the time, and I do have bits and pieces of material. But it's something we'd really like to develop. It'd be quite easy to knock out eight to ten in-character songs and release it as quickly as possible to capitalise on the popularity of the group, but it would feel entirely wrong." In 2011, Godflesh (along with Black Sabbath, Napalm Death, Judas Priest and Led Zeppelin) were recognised by the UK-based Home of Metal archival project as significant contributors to the heavy metal genre. With this accolade absorbed, Broadrick continued to confirm the existence of a new Godflesh album throughout 2012, saying it would most likely be out in 2013 after a new EP. In 2013, Godflesh released their first new recording in over twelve years, a cover of Slaughter's "F.O.D. (Fuck of Death)", through Decibel magazine's flexi disc Series. The flexi disc was included in the November 2013 issue of Decibel. Later in 2013, Godflesh performed Pure in its entirety at Roadburn, featuring Hampson on guitar for part of the set. In 2014, Broadrick said that disbanding Godflesh was the best thing they had ever done and stated that the upcoming album was his favourite since 1994's Selfless.

### A World Lit Only by Fire and Post Self (2014–2021)

From 2012 to 2014, Godflesh were in Broadrick's private studio, working on their seventh album. The process was slow due to Broadrick's commitment to making a proper Godflesh album instead of a hollow exploitation of the reformation's media attention. After a long revision process, the 2014 EP Decline & Fall and the 2014 album A World Lit Only by Fire came out of those sessions, both of which featured Broadrick on an eight-string guitar. The album was the band's first to chart in the U.S., and it was met with critical acclaim, drawing praise for its extreme weight and being lauded as the ideal return for Godflesh. Musically, Broadrick stated A World Lit Only by Fire was most similar to Streetcleaner and Pure, and Green added that it was "spiritually" close to those first few releases. About the album's sound, Broadrick said, "It's extremely minimal and very, very brutal — a fairly unforgiving record, really, I think." Decline & Fall and A World Lit Only by Fire were heavy, distorted, riff-driven industrial releases that Broadrick saw as a return to the direct simplicity of Godflesh's original form. All of the recording, engineering, artwork and packaging was done by Godflesh through Broadrick's own label, Avalanche Recordings.

Following a number of tours in support of A World Lit Only by Fire and a failed attempt at an In Dub remix release with Parsons providing additional drumming, Godflesh returned to the studio in 2016 for a new album. Post Self, the band's eighth studio album, was released on 17 November 2017, shortly after another performance of Streetcleaner in its entirety. Unlike A World Lit Only by Fire's focus on making a bombastic comeback, Post Self proved more introspective and moody. Most of the traditional metal riffs were dropped in favour of atmosphere, noise and experimentation. Post Self received similar high praise as its predecessor, and both appeared on several publications' year-end lists. After the release of Post Self, Broadrick avoided interviews, hoping to give listeners time to form their own opinions on the music and to retain some of the album's "mystique". Godflesh embarked on a few brief tours and played scattered festivals, including the 2018 edition of Roadburn where they performed all of Selfless live for the first time. Broadrick spent a month reprogramming the album's percussion from scratch since the original drum parts no longer existed. In an interview with French magazine New Noise, Broadrick said that Selfless was the last Godflesh album he wanted to play in its entirety.

In July 2021, Godflesh announced a compilation album entitled Long Live the New Flesh, which featured almost all of the band's studio material released since reformation in 2010. A shortened digital edition, New Flesh in Dub Vol 1, comprising most of Godflesh's reformation-era remixes alongside two otherwise unreleased tracks from the Post Self sessions, was released ahead of the full compilation.

### Purge (2022–present)

In January 2022, Broadrick confirmed that a new Godflesh album was being demoed for release later that year and that two live albums documenting the performances of Pure and Selfless at Roadburn 2013 and 2018 respectively were forthcoming. The former of the pair, titled Pure : Live, was released on 2 November 2022. On 1 March 2023, the band revealed that Purge, their ninth studio album, would release on 9 June 2023. It was preceded by the single "Nero" on 3 April along with an announcement of a North American tour and that New Flesh in Dub Vol 2 was planned for release in mid-2024.

## Style and influence

### Musical style and legacy

Drawing inspiration from early industrial and noise groups such as Throbbing Gristle and Whitehouse, as well as from ambient musician Brian Eno, noise rock band Swans, post-punk innovators Killing Joke and also Siouxsie and the Banshees and Public Image Ltd., fellow Birmingham heavy metal band Black Sabbath and a wide range of hip hop artists such as Public Enemy, Eric B. & Rakim and Run-DMC, Godflesh were among the pioneers of both industrial metal and post-metal and are considered a significant experimental metal and extreme metal band. Broadrick and Green have both distanced themselves from the industrial metal title, though they concede that the label is accurate on a literal level. They consider Godflesh to be a continuation of post-punk, specifically in regard to furthering Killing Joke's sound. When asked in 2005 how he felt about being one of industrial metal's founders, Broadrick said, "I think I accidentally formed this genre. I don't even really know what this genre is; I just make music." He expanded upon this statement in 2018 by calling the term industrial metal "limiting".

Godflesh's sound is characterised by a combination of programmed drum machine beats, percussive bass and distorted heavy metal guitar. Though the band would later employ human drumming for Songs of Love and Hate and Hymns in 1996 and 2001 respectively (a decision that Broadrick believed compromised the group's musical identity), Godflesh's early sound was dominated by artificial, stiff drum loops and an uncommon focus on bass. On those early influential releases, the rhythms, synths and samples were credited to "Machine" or "Machines" despite being done by Broadrick. Initially, the employment of mechanical percussion was done out of necessity. When Broadrick rejoined Fall of Because with Green to form Godflesh, the beats he desired were too difficult to play acoustically, so he instead used an Alesis HR-16 drum machine; this decision would prove defining for both Godflesh and for industrial metal at large. Godflesh's writing and recording process is guided by Broadrick and generally happens without the aid of a professional studio or producer. According to Broadrick, "50% of the Godflesh material was often born from rhythms and grooves", and he called the hands-on recording process "primitive".

About the band's tuning, Green said in a 1997 interview, "We would just tune down as low as we could. We couldn't hear anything! But eventually we had to pick a pitch to tune to, so we chose C-sharp. It's low and heavy, but you can still make out the notes." For the early Godflesh albums, the guitar tunings are tuned down to Drop C-sharp, B, and A. Starting with 2014's A World Lit Only by Fire, Broadrick started using an 8-string guitar tuned to F sharp.

Mike Patton of bands Faith No More and Mr. Bungle was a fan of Godflesh's sound, calling them "ridiculously heavy" and "one of the few bands that can make you wonder whether or not your batteries are running out on your stereo." Patton invited Broadrick to join Faith No More as guitarist after Jim Martin left the band in 1993. He was also asked to join Danzig shortly after Godflesh toured the U.S. with them and Type O Negative in 1994. In both instances, Broadrick declined the offer in favour of continuing to reside in the United Kingdom and work on his own music. Other artists such as Metallica, Fear Factory, Korn, Isis, Neurosis, Pelican, Prurient, Helmet, Prong, Nailbomb, Code Orange, Mortiis, Devin Townsend and Converge were inspired by Godflesh, and in 2014 MetalSucks wrote, "Everyone from Nine Inch Nails to Fear Factory to Batillus probably owes these dudes a couple of checks". Despite acclaim among critics and fellow musicians, Godflesh have only experienced minimal commercial success; in 2002, shortly before the band dissolved, Broadrick said he had no illusions about selling out stadiums. The goal of Godflesh, according to him, was simply to put out good albums.

### Visual style

Being interested in horror and art house cinema, both Broadrick and Green incorporated many references to such films in Godflesh. The image on the front of 1988's Godflesh EP is a still from the 1966 John Frankenheimer film Seconds. Streetcleaner's cover is an image from the movie Altered States, a 1980 horror film by director Ken Russell, and the album's liner notes feature frames from David Lynch's Eraserhead (1977). The cover of Merciless is derived from the 1943 experimental film Meshes of the Afternoon, by Maya Deren. Broadrick repeatedly cited another Russell movie, The Devils (1971), as an influence of Godflesh's sound. Additionally, Godflesh appeared in the 1995 film Hideaway; during one of the movie's club scenes, Broadrick and Green can be seen playing onstage in the background, performing the song "Nihil" from Cold World.

Apart from cinema, Christian iconography helped inform Godflesh's visual style. Broadrick, originally anti-religious, found churches and their related imagery intimidating. In a 2006 interview, Broadrick said, "The imagery of religion, the feeling of like when you walk into a cathedral, the huge feeling of intimidation that you get from Christian religion – everything to do with religion I find totally obsessive." Broadrick described the worship setting as "suffocating" and "claustrophobic", saying it played a large role in forming Godflesh's style. The covers of Streetcleaner, Songs of Love and Hate, Love and Hate in Dub and A World Lit Only by Fire showcase such Christian imagery, as do Godflesh's live performances.

### Vocal style and lyrical themes

Early Godflesh vocals came in the form of screaming and death growls, and lyrics were scarce. Over time, Broadrick's delivery expanded to include singing, softness and moments of melody, all things that he would further explore with Jesu. Songs of Love and Hate and Hymns saw Godflesh's greatest profusion of vocals, while A World Lit Only by Fire and Post Self returned to Broadrick's predominantly succinct style. Broadrick's voice has been compared to Killing Joke singer Jaz Coleman's, a self-proclaimed vocal influence in addition to Kelvin Morris from Discharge and Michael Gira's early work with Swans. Godflesh lyrics are cryptic, bleak and generally sparse. Many of Godflesh's themes deal with inner conflict, violence, destruction, corruption, religion, loss, emotional extremes and fear. In 2012, Broadrick stated that singing was "a necessary evil" to him and that he never felt he could do it properly; the role of vocalist came to him simply because he was bold enough to approach the microphone.

In a 1990 review of the band's self-titled EP, Simon Reynolds of Melody Maker noted that Godflesh's music rebelled against the overt masculinity found in most metal; he wrote, "Rather than feminise themselves, they'd rather their masculinity was defeated, their strong bodies crushed and pulverised". Broadrick echoed this sentiment in the same year, insulting the "celebration of male ego that comes with most metal" and distancing Godflesh from masculine expression; he reinforced his stance in 2012, saying, "Godflesh isn't that all-conquering male thing, on stage with a fucking sword and saying they're going to 'rape and pillage'. Pure comedy." Throughout the group's career, Broadrick continued to emphasise the defensive quality of the music over its offensive quality, citing aggression as a shield against the traumas of the world.

Broadrick has taken significant lyrical and thematic inspiration from Leonard Cohen. Both artists have an album titled Songs of Love and Hate, and the Godflesh track "Avalanche Master Song" combines two Cohen song titles. In the song "Mothra" (from Pure), Godflesh borrows the lyrics "Your pain is no credential here / It's just the shadow of my wound" from the song "Avalanche" on the aforementioned Cohen album.

## Live performances

Godflesh typically perform as a duo, with Green playing bass and Broadrick providing both guitar and vocals while the percussion is relayed over the speakers, often at a notably high volume. This lineup, however, has not been constant over the band's career. In concerts supporting Streetcleaner, Neville occasionally came onstage and played second guitar, and Hampson did the same for Pure. For the tours in support of Selfless, Songs of Love and Hate and Hymns, the band played with a live drummer, though Broadrick believed this incorporation to be a misstep. In a 2014 interview, he clarified that the percussive additions of Mantia and Parsons brought "something amazing" to Godflesh, but that the band's name should have changed so as to not dilute the original focus on machine drums. A handful of other musicians (namely, Mick Harris in 1991; Diarmuid Dalton in 1997, 1999 and 2001; Steve Hough in 1997 and 1999 and both Paul Raven and Jaz Coleman of Killing Joke in 2002) have briefly performed with Godflesh.

The band's stage show is, like their music, minimal and focused. Godflesh only occasionally employ a smoke machine, and the lights are generally static; Toby Cook of The Quietus highlighted this unusual approach in a review of a 2014 concert, writing, "Over-lit and with no dry ice in sight, they look alarmingly exposed". Images of industrial urban sprawls, Christian iconography and apocalyptic landscapes cycle through the projector. The music itself is presented as noisy and overwhelming, with some critics calling the sound even more cacophonous and dissonant than the band's studio work. Gaps between songs are normally filled with feedback, and the band rarely speak to the crowd. In response to a question about Godflesh's laconic stage presence, Broadrick said, "I don't communicate with people. We just make music. We come on stage and I don't say anything to anyone. And that alone made people really mad. I find it really bizarre. I can't believe that people would disown a band because I'm not telling them how much I'm going to kick their asses tonight."

In a 2011 interview, Broadrick called playing live "a necessary evil" that is "so rarely right and so frequently wrong", but at the same time said that Godflesh "is probably even more so a live band than it is a recorded band" because those imperfections only work to accentuate the grinding chaos of the music (which is something that critics have noticed, too). Broadrick went on to say that it was the "rage" of Godflesh that made it work in a live setting.

## Members

### Current lineup

- Justin Broadrick – guitars, vocals, programming (1988–2002, 2009–present), drums (1984–1987)
- B. C. Green – bass, programming (1982–1987, 1988–2001, 2009–present)

### Former members

- Paul Neville – guitars (1982–1987, 1989–1991)
- Robert Hampson – guitars (1991–1992)
- Bryan Mantia – drums (1994–1996)
- Ted Parsons – drums (1996–2002)
- Paul Raven – bass (2002; died 2007)

### Former touring musicians

- Mick Harris – drums (1991)
- Steve Hough – guitars (1997, 1999)
- Diarmuid Dalton – keyboards (1997, 1999, 2001)

### Timeline

## Discography

- Streetcleaner (1989)
- Pure (1992)
- Selfless (1994)
- Songs of Love and Hate (1996)
- Us and Them (1999)
- Hymns (2001)
- A World Lit Only by Fire (2014)
- Post Self (2017)
- Purge'' (2023)

English industrial metal band

Musical groups from Birmingham, West Midlands


The Revenant (2015 film)

The Revenant is a 2015 American Western action drama film directed by Alejandro G. Iñárritu. The screenplay by Mark L. Smith and Iñárritu is based in part on Michael Punke's 2002 novel The Revenant, which describes frontiersman Hugh Glass's experiences in 1823, and which is based on the 1915 poem The Song of Hugh Glass. The film stars Leonardo DiCaprio and Tom Hardy.

In August 2001, Akiva Goldsman purchased Punke's manuscript. Iñárritu signed on to direct The Revenant in August 2011; in April 2014, after several delays due to other projects, Iñárritu confirmed that he was beginning work on it and that DiCaprio had the lead role. Principal photography began in October 2014. Location and crew concerns delayed production from May to August 2015.

The Revenant premiered at the TCL Chinese Theatre in Los Angeles, California on December 16, 2015. It had a limited release on December 25 and a wide release on January 8, 2016. It was a blockbuster, grossing \$533 million worldwide. It received critical acclaim, with praise for the performances, particularly for DiCaprio and Hardy, Iñárritu's direction, and Lubezki's cinematography; however, some criticism went to its runtime.

It won three Golden Globe Awards and five BAFTA Awards, including Best Film at both ceremonies. At the 88th Academy Awards, the film received 12 nominations, including Best Picture and Best Supporting Actor (Hardy). It won the Academy Awards for Best Director (Iñárritu, his second consecutive in that category), Best Actor (DiCaprio, his first after 5 previous nominations), and Best Cinematography (Emmanuel Lubezki, his third consecutive in that category). DiCaprio also won the Golden Globe Award, the Screen Actors Guild Award, the BAFTA Award, and the Critics' Choice Award for Best Actor.

## Plot

In late 1823, fur trapper Hugh Glass guides Captain Andrew Henry's trappers through the territory of the present-day Dakotas. While he and his half-Pawnee son, Hawk, are hunting, the company's camp is attacked by an Arikara war party which is seeking to recover its Chief's abducted daughter, Powaqa. Many of the trappers are killed during the fight, and the rest of them escape onto a boat. Guided by Glass, the survivors travel to Fort Kiowa on foot because Glass believes that traveling downriver will make them vulnerable. After docking, the crew stashes the pelts near the shore.

While he is scouting game, Glass is mauled and left near death by a female grizzly bear which is guarding its cubs. Trapper John Fitzgerald, fearing another Arikara attack, argues that the group must mercy-kill Glass and keep moving. Henry agrees, but he is unable to pull the trigger. Instead, he offers money in order to pay someone to stay with Glass and bury him after he dies. When the only volunteers are Hawk and the young Jim Bridger, Fitzgerald agrees to stay for money, in order to recoup his losses from the abandoned pelts.

After the others leave, Fitzgerald attempts to smother Glass but he is neutralized by Hawk. Fitzgerald, afraid that Hawk's loud reaction to his attempt to kill Glass could alert Bridger, who is away gathering water, stabs Hawk to death as Glass watches helplessly. The next morning, Fitzgerald convinces Bridger, who is unaware of Hawk's murder, that the Arikara are approaching and they must abandon Glass. At first, Bridger protests, but he ultimately follows Fitzgerald after the latter leaves Glass half-buried alive in a makeshift grave. Bridger leaves his canteen, in which he engraved a spiral symbol, with Glass. After they depart, Fitzgerald admits that he lied about the approaching Arikara. When Fitzgerald and Bridger later meet Henry at the fort, Fitzgerald tells Henry that Glass died and Hawk vanished. Bridger is complicit in the lie about Glass's death, but he knows nothing about Hawk's murder.

Glass begins his arduous journey through the wilderness. He performs a crude cauterization of his wounds and eludes the pursuing Arikara by jumping into water rapids. He later encounters Pawnee refugee Hikuc, who says that "revenge is in the Creator's hands." The men share bison meat and travel. As a storm approaches, Hikuc constructs a makeshift sweat lodge for a feverish Glass to shelter in. After a hallucinogenic experience in the lodge, Glass emerges to discover that his wounds are healing, but French hunters have lynched Hikuc. He infiltrates their camp and sees the leader raping Powaqa. Glass frees her, and while she castrates her rapist, he kills several hunters, and recovers Hikuc's horse. The following day, Glass is ambushed and driven over a cliff on his horse by the Arikara. He survives the stormy night by eviscerating the dead horse and sheltering inside its carcass.

A frightened French survivor staggers into Fort Kiowa, and Bridger recognizes his spiral engraved canteen as Glass's. Believing that it might be a surviving Hawk, Henry organizes a search party. Realizing that Glass is alive, Fitzgerald empties the outpost's safe and flees. The search party finds the exhausted Glass. Furious, Henry orders the arrest of Bridger, but Glass vouches for Bridger by stating that he was not present when Fitzgerald murdered Hawk and he was later deceived and threatened by the higher-ranking Fitzgerald. Glass and Henry set out in pursuit of Fitzgerald.

After the two men split up, Fitzgerald ambushes, kills, and scalps Henry. Glass finds Henry's corpse, places it on his horse in an attempt to act as a decoy, and shoots Fitzgerald in the arm. He pursues Fitzgerald to a riverbank, where they engage in a brutal fight. Glass is about to kill Fitzgerald, but he spots a band of Arikara downstream. He remembers Hikuc's words and pushes Fitzgerald downstream into the hands of the Arikara. Their chief Elk Dog kills and scalps Fitzgerald, and the Arikara (who have found Powaqa) spare Glass. Glass retreats into the mountains, where he is visited by his wife's spirit.

## Cast

- Leonardo DiCaprio as Hugh Glass
- Tom Hardy as John S. Fitzgerald
- Domhnall Gleeson as Andrew Henry
- Will Poulter as Jim Bridger
- Forrest Goodluck as Hawk
- Paul Anderson as Anderson
- Kristoffer Joner as Murphy
- Duane Howard as Elk Dog
- Melaw Nakehk'o as Powaqa
- Arthur Redcloud as Hikuc
- Lukas Haas as Jones
- Brendan Fletcher as Fryman
- Tom Guiry as Billy Brother Trapper
- Grace Dove as Hugh Glass's wife

## Production

### Development

Development of The Revenant began in August 2001, with producer Akiva Goldsman acquiring the rights to Michael Punke's then-unpublished manuscript. David Rabe had written the film's script. The production was picked up by Park Chan-wook, with Samuel L. Jackson in mind to star. Park later left the project. The development stalled until 2010 when Mark L. Smith wrote a new adaptation of the novel for Steve Golin's Anonymous Content. In May 2010, Smith revealed that John Hillcoat was attached to direct the film and that Christian Bale was in negotiation to star. Hillcoat left the project in October 2010. Jean-François Richet was considered to replace him, but Alejandro G. Iñárritu signed on to direct in August 2011. Goldsman was also confirmed to be producing with Weed Road Pictures. In November, New Regency Productions joined to produce with Anonymous Content, and 20th Century Fox was confirmed to be distributing the film. Days later, Iñárritu stated that he was seeking Leonardo DiCaprio and Sean Penn for the two lead roles.

Once Iñárritu agreed to direct, he began working with Smith on script rewrites. In an interview with Creative Screenwriting, Smith admitted that during this process, he was unsure if Iñárritu would even be able to film some of the sequences they wrote. He recalled, "He would have some ideas and I would say, 'Alejandro, we can't pull this off. It's not going to work', and he would say, 'Mark, trust me, we can do this.' In the end, he was right."

The project was put on hold in March 2012, as New Regency hired Iñárritu to direct an adaptation of Flim-Flam Man, Jennifer Vogel's non-fiction book about her criminal father. Penn was also under consideration for the lead role in that film. In December 2012, Iñárritu announced that his next film would be Birdman or (The Unexpected Virtue of Ignorance), a comedy-drama about an actor who once played a famous superhero. He ended up winning the Academy Awards for Best Director and Best Original Screenplay, with the film winning Best Picture. Filming took place in March 2013. The Revenant was scheduled to begin production right after Birdman wrapped.

A production budget of \$60 million was granted, with \$30 million funded by New Regency. Brett Ratner's RatPac-Dune Entertainment, a joint venture between Ratner's RatPac Entertainment and 20th Century Fox's former financing partner, Dune Entertainment, also funded the film. Worldview Entertainment, which also co-financed Birdman, was originally set to fund the film but backed out in July 2014 due to the departure of its CEO Christopher Woodrow. New Regency approached 20th Century Fox for additional funding, but the company declined, citing the pay-or-play contracts made for both DiCaprio and Tom Hardy, which would require that the actors be paid regardless of whether the film is completed. Annapurna Pictures' Megan Ellison entered negotiations to finance the film shortly after. The Chinese company Guangdong Alpha Animation and Culture Company partially financed the film.

### Filming

Principal photography for The Revenant began in October 2014. A planned two-week break from filming in December was extended to six weeks which forced Tom Hardy to drop out of Suicide Squad. In February 2015, the director, who shot the film using natural lighting, stated that production would last "until the end of April or May", as the crew is "shooting in such remote far-away locations that, by the time we arrive and have to return, we have already spent 40% of the day". Ultimately, principal photography wrapped in August 2015.

The waterfall scenes were filmed at the Kootenai Falls near Libby, Montana. Though the initial plan was to film the last scenes in Canada, the weather was ultimately too warm, leading the filmmakers to locations near the Rio Olivia at the tip of Argentina with snow on the ground, to shoot the ending.

Crew members often complained about difficult filming, with many quitting or being fired. Mary Parent was then brought in as a producer. Iñárritu stated that some of the crew members had left the production, explaining that "as a director, if I identify a violin that is out of tune, I have to take that from the orchestra." On his experience, DiCaprio stated: "I can name 30 or 40 sequences that were some of the most difficult things I've ever had to do. Whether it's going in and out of frozen rivers, or sleeping in animal carcasses, or what I ate on set. [I was] enduring freezing cold and possible hypothermia constantly."

The director had stated that he originally wanted to film chronologically, a process that would have added \$7 million to the production budget. Later he confirmed that it was shot in sequence, though Hardy had stated that it was impossible due to weather conditions.

In July 2015, it was reported that the budget had ballooned from the original \$60 million to \$95 million, and by the time production wrapped it had reached \$135 million.

### Visual effects

The visual effects were produced primarily by Industrial Light & Magic (ILM). Other companies, such as Moving Picture Company (MPC) and Cinesite, also created visual effects.

### Music

The musical score was composed by Japanese musician Ryuichi Sakamoto and German electronic musician Alva Noto with additional music composed by Bryce Dessner. The main body of the score was recorded at the Seattlemusic Scoring Stage in the Bastyr Chapel in greater Seattle, Washington by musicians of the Northwest Sinfonia. Sakamoto conducted these sessions. Bryce Dessner's portion of the score was performed by the 25-piece Berlin-based orchestra known as "s t a r g a z e" under conductor André de Ridder. Additional licensed music includes "Become Ocean", the Pulitzer Prize and Grammy Award-winning work of John Luther Adams as recorded by the Seattle Symphony with conductor Ludovic Morlot and an excerpt of "Jetsun Mila" from French musician and composer Eliane Radigue. A soundtrack album was released online on December 25, 2015, and on CD on January 8, 2016. Milan Records released a vinyl pressing of the soundtrack in April 2016.

The score by Sakamoto and Noto was ruled ineligible for the Academy Award for Best Original Score at the 2016 Oscars as it was deemed that it was "assembled from the music of more than one composer".

### Documentary

The movie was accompanied by a 44-minute documentary, named A World Unseen, highlighting the process of making the production. A World Unseen was released on January 21, 2016, on YouTube. Eliot Rausch served as its director.

## Themes

In the Los Angeles Review of Books, critic Wai Chee Dimock compared The Revenant's themes with those addressed in the literary works of James Fenimore Cooper, particularly The Last of the Mohicans. Dimock argues that the film reinterprets the concept of "half-breeds" from a derogatory idea that Cooper despised to an aesthetic way in which to see the world. She compared both works' protagonists—Glass and Hawk-eye—as literary foils, with Glass living an inversion of the latter's biography and perspective.

In the documentary A World Unseen, the director stated that for The Revenant's main themes, he revisits the issues and concerns of intense parental and filial relations, which are readily recognizable as a recurrent theme from his previous works. Regarding the theme of revenge, he mentioned that the approach of vengeance in the film needs to be significantly tempered by anyone who would want to see vengeance as either an effective or useful moral to be applied in life. In the end, he said, there can only be a disappointment and lack of fulfillment for anyone who looks to revenge as providing a higher purpose for living or a life-defining purpose.

## Historical accuracy

The Guardian reported, "The backstory about Glass's love for a Pawnee woman is fiction. It has been suggested the real Glass had such a relationship, but there's no firm evidence—and no evidence that he had any children. ... As for the ending, it has been changed in one significant way: in real life, nobody got killed."

Canadian actor Roy Dupuis was strongly critical of the portrayal of French-Canadian voyageurs as murderous rapists. Dupuis was originally offered a role as a voyageur, but he rejected it due to perceptions of anti-French bias and historical inaccuracies. According to Allan Greer, the Canada Research Chair of Colonial North America, "generally the American traders had a worse reputation than the Canadians."

The filmmakers made a special point of emphasizing the importance of historical issues of ethnicity approached in the movie and represented in the mixed ethnic background of Hugh Glass's son portrayed in the film (portrayed as half Pawnee by Glass's wife) as relating to his own life and his identification with ethnic concerns. The director has referred to having encountered constant xenophobia and stated that: "These constant and relentless xenophobic (comments) have been widely spread by the media without shame, embraced and cheered by leaders and communities around the US. The foundation of all this is so outrageous that it can easily be minimized as an SNL sketch, a mere entertainment, a joke ... I debated with myself, if I should bring up this uncomfortable subject tonight but in light of the constant and relentless xenophobic comments that have been expressed recently against my Mexican fellows, it is inevitable."

Bruce Bradley of True West Magazine pointed out several anachronisms hinted in the film such as John Fitzgerald talking about his father having friends in the Texas Rangers and about going to Texas and joining the army himself. However, in 1823, Texas was still part of Mexico and therefore would not have had an American army, let alone the Texas Rangers. Another example being Hugh Glass discovering a mountain of buffalo skulls while working his way to Fort Kiowa despite the buffalo slaughter not occurring until the 1870s.

### Native American culture

To portray Arikara culture accurately, the filmmakers hired several cultural consultants and teamed up with two linguists to provide faithful Arikara and Pawnee language lines for actors. Hikuc, the Pawnee man who helps Glass survive, is played by a Navajo actor. However, in one scene, a Pawnee character rescuing Glass is accompanied by a voiceover in Inupiaq, which is spoken in Arctic Alaska, thousands of kilometres away and a different language family from Pawnee. The voiceover was a recording of Doreen Nutaaq Simmonds reading a poem from a John Luther Adams recording; the words originally came from an Inuit woman named Uvavnuk, an angakkuq (shaman) and oral poet.

## Release

The Revenant had a limited release in the United States on December 25, 2015, including Los Angeles—making it eligible for the 88th Academy Awards—before being released nationwide on January 8, 2016. The film opened in Australia on January 7, 2016, and in the United Kingdom on January 15, 2016. In the Philippines, the film's release date was originally set for January 27, 2016, but it was eventually delayed one week to February 3, 2016. Although studios initially chose not to pursue a theatrical release in China, following its three wins at the 88th Academy Awards on February 28, 2016, the film was granted a release in China but with several cuts. It was released on March 18, 2016.

### Home media

The DVD, Blu-ray and 4K Ultra HD Blu-ray were released on April 12, 2016, in the US. Opening sales of the DVD along with online streaming orders placed The Revenant as number one in sales at Amazon. Distribution to major rental outlets in the US was done on May 17, 2016.

### Piracy

On December 20, 2015, less than one week before its release, screener copies of The Revenant were leaked online. The FBI linked this to Alcon Entertainment CEO Andrew Kosove, who denied responsibility. In October 2016, a former 20th Century Fox employee was fined \$1.12 million in a separate case for uploading both The Revenant and The Peanuts Movie online.

## Reception

### Box office

The Revenant grossed \$183.6 million in the United States and Canada and \$349.3 million in other countries for a worldwide total of \$533 million, against a production budget of \$135 million. In March 2016, before the film had completed its theatrical domestic and international runs, Deadline Hollywood calculated the net profit of the film to be \$61.6 million when factoring together all expenses and revenues.

In North America, The Revenant opened in limited release on December 25, 2015, and over the weekend grossed \$474,560 from four theaters in New York City and Los Angeles (\$118,640 per screen), finishing twenty-third at the box office. It was the second-biggest theater average of 2015 behind the \$130,000 four-screen debut of Steve Jobs. The film earned a total of \$1.6 million from its two-week limited run before expanding wide on January 8, 2016, across 3,371 theaters.

It earned \$2.3 million from its early Thursday preview showings from 2,510 theaters. On its opening day, the film earned \$14.4 million, ranking first at the box office. The film grossed \$39.8 million in its opening weekend from 3,375 theaters, exceeding initial projections by 70%, and finishing second at the box office behind Star Wars: The Force Awakens (\$42.4 million), which was on its fourth weekend of play. It was the director's biggest opening of all time, and the fourth-biggest for DiCaprio and supporting actor Tom Hardy. Critics noted that The Force Awakens had an advantage from playing at 781 more theaters, family-friendly Sunday matinees, and all North American IMAX theaters. Nevertheless, The Revenant played very balanced across the US and overperformed in all states except the Northeast region. Its wide release weekend is among the top openings in January. It topped the box office in its fifth weekend overall and third weekend in wide release after competing with Ride Along 2 in its second weekend. It added \$16 million in its third weekend, which was down 49.7% but topped the box office, despite a blizzard blanketing most of the East Coast which reportedly hurt many films' box office performance. The following weekend it was overtaken by Fox's own animated movie Kung Fu Panda 3 thereby topping the box office for one weekend. Following the announcement of the Oscar nominees on January 14, The Revenant witnessed the biggest boost among the Best Picture category, jumping from \$54.1 million to \$170.5 million, an increase of +215% up to the Oscar ceremony in the weekend ending February 28.

Outside North America, the film secured a release in 78 countries. It made \$20.5 million from 2,407 screens in just 18 markets, placing behind The Force Awakens at the international box office chart and first among newly released films. The following weekend, it added \$32.3 million from 25 markets on 4,849 screens. The film topped the international box office in its third weekend—the same weekend when it topped the US box office—overtaking The Force Awakens with \$33.7 million from 48 markets. In the United Kingdom and Ireland, it took the number 1 spot with \$7.87 million or £5.2 million (\$7.4 million) from 589 theaters and remained there for a second weekend declining by 24% with £3.86 million (\$5.5 million), and for a third weekend. Similarly, in Russia, it passed The Force Awakens to take the top spot with \$7.5 million from 1,063 screens. In France, it has the biggest opening day in Paris and the third-biggest opening weekend of 2016 with \$8.2 million. It also opened at No. 1 in Mexico (\$5.1 million), Spain (\$4 million), the Netherlands (\$1.3 million), Belgium (\$1.1 million), Argentina (\$955,000), Sweden (\$914,000), South Korea, Denmark, Norway, Israel, Egypt and Portugal among other markets. In Germany (\$4.6 million) and Australia (\$2.9 million), it debuted at No. 2 both behind The Force Awakens and in Brazil (\$2.17 million) behind The Ten Commandments. It had one of the top ten openings of all time for a Fox film, not accounting for inflation in South Korea with \$5.7 million and went on to top the box office there for a second weekend with \$3.22 million despite cold weather affecting theater attendance resulting in low box office performance. In Russia, though not opening at number 1, it topped the box office in its second weekend with \$4.4 million—more than The Force Awakens—and went on to the top for a third weekend with \$3.6 million. In China, it had an opening day of around \$11 million from more than 11,000 screens, including \$250,000 in midnight previews, and \$23 million in two days. In its opening weekend, it grossed \$31 million, coming in second place behind the animated Zootopia. IMAX comprised \$2.3 million on 278 screens. In terms of total earnings, its largest markets outside of the US and Canada are China (\$58.6 million), the United Kingdom (\$32.8 million), Germany (\$28.7 million) and France (\$28.2 million). The film opened in Japan on March 23.

### Critical response

On Rotten Tomatoes, the film has an approval rating of 78%, based on 399 reviews, with an average rating of 7.80/10. The website's critical consensus reads: "As starkly beautiful as it is harshly uncompromising, The Revenant uses Leonardo DiCaprio's committed performance as fuel for an absorbing drama that offers punishing challenges—and rich rewards." On Metacritic, the film has a weighted average score of 76 out of 100, based on 50 critics, indicating "generally favorable reviews". Audiences polled by CinemaScore gave the film an average grade of "B+" on an A+ to F scale, and PostTrak reported audiences gave it an overall positive score of 85% and a 59% "definite recommend".

Reviewers cited in a CBS News survey of critics highly praised DiCaprio's performance, referring to it as an "astonishing testament to his commitment to a role" and as an "anchoring performance of ferocious 200 percent commitment". Peter Travers of Rolling Stone called DiCaprio's acting "a virtuoso performance, thrilling in its brute force and silent eloquence". Writing for NY Magazine/Vulture, David Edelstein called the film a "tour de force" and "[b]leak as hell but considerably more beautiful", but noted it had "traditional masculinity instead of a search for what illuminates man's inhumanity to man".

Justin Chang of Variety wrote Iñárritu "increasingly succumbs to the air of grim overdetermination that has marred much of [his] past work" and it was "an imposing vision... but also an inflated and emotionally stunted one". Stephanie Zacharek, writing for Time magazine, gave a positive review to the film stating: "Inarritu may have fashioned The Revenant as the ultimate endurance test, but as Glass, DiCaprio simply endures. He gives the movie a beating heart, offering it up, figuratively speaking, alive and bloody on a platter. It—he—is the most visceral effect in the movie: revenge served warm. Bon appetit." Richard Brody of The New Yorker was critical of the film, and said that Emmanuel Lubezki's images were mere "pictorial ornament[s] to Alejandro G. Iñárritu's bland theatrical stagings". A critic for Santa Barbara Independent considered it a remake of Man in the Wilderness (1971).

Slant Magazine's writer Ed Gonzalez suggested that the Slant staff mostly disliked the film: "Our contempt for The Revenant knows no limits." Gonzalez unfavorably compared Iñárritu's work to Terrence Malick's 2005 film The New World. In the official review, Slant writer Jaime N. Christley wrote: "The Revenant [is] a misery-fest that plants its narrative flags as carelessly as a Roland Emmerich blockbuster, guaranteeing us a viewing experience almost as arduous as the trials depicted on screen, before reaching a conclusion that's sealed the moment audiences first meet the key players. After an obligatory false calm, The Revenant's proper opening scene is a show-stopping massacre at a fur trapper's campsite. It's the kind of thing Howard Hawks would have handled—and did, in The Big Sky—in under 90 seconds, with mostly off-camera particulars and minimal effects, but Iñárritu forces it to resemble the Normandy Beach sequence in Saving Private Ryan as much as history or sense will allow, and then some." The Revenant was ranked 22nd on Metacritic's and 79th on Rotten Tomatoes' list of best films of 2015.

### Accolades

The Revenant has received numerous award nominations and wins, particularly for DiCaprio's performance, Iñárritu's direction and Lubezki's cinematography. At the 88th Annual Academy Awards, Iñárritu won Best Director for the second consecutive time, Emmanuel Lubezki won Best Cinematography for the third consecutive time, and DiCaprio won his first award for Best Actor. Hardy lost the award for Best Supporting Actor to Mark Rylance for Bridge of Spies, and the film itself lost Best Picture to Spotlight.

At the 73rd Golden Globe Awards, it won three awards: Best Motion Picture – Drama, Best Director, and Best Actor – Drama. It also had a nomination for Best Original Score. On January 14, 2016, the film received 12 Academy Award nominations (more than any other film at the ceremony), including Best Picture and Best Director, as well as Best Actor and Best Supporting Actor. On February 14, 2016, the film received the most awards at 69th British Academy Film Awards out of eight nominations, with five, including Best Film, Best Director, Best Cinematography, Best Sound, and Best Leading Actor. It received nine Critics' Choice Awards nominations, winning two Best Actor and Best Cinematography.

Tom Hardy won the Best British Actor award at the London Film Critics' Circle and was runner-up for Best Supporting Actor at Dallas–Fort Worth Film Critics Association. DiCaprio was awarded Outstanding Performance by a Male Actor in a Leading Role award at 22nd Screen Actors Guild Awards, and Iñárritu received Outstanding Directing – Feature Film award at 68th Directors Guild of America Awards. It received five Satellite Awards nominations, winning the award of Best Actor for DiCaprio.

On May 2, 2016, Time magazine included both DiCaprio and Iñárritu in its issue of the 100 Most Influential People of 2015, with a cover photograph of DiCaprio on the magazine. John Kerry, then the US Secretary of State, wrote a short testimonial to DiCaprio for this issue of Time stating that DiCaprio's dedication drives him to succeed and "that's how he takes himself back 200 years to create an Oscar-winning, bear-brawling, powerhouse performance in The Revenant."

## See also

- Survival film
- Man in the Wilderness, a 1971 Western film loosely based on the Hugh Glass story
- Lord Grizzly, a 1954 biographical novel by Frederick Manfred, about the Hugh Glass story

2015 American film by Alejandro González Iñárritu

Films directed by Alejandro González Iñárritu

Films featuring a Best Actor Academy Award-winning performance

Films featuring a Best Drama Actor Golden Globe winning performance

Films whose cinematographer won the Best Cinematography Academy Award

Films whose director won the Best Directing Academy Award

Films whose director won the Best Direction BAFTA Award

Films whose director won the Best Director Golden Globe

Films with screenplays by Alejandro González Iñárritu

Golden Eagle Award (Russia) for Best Foreign Language Film winners

Western (genre) films based on actual events


Shapley–Folkman lemma

The Shapley–Folkman lemma is a result in convex geometry that describes the Minkowski addition of sets in a vector space. It is named after mathematicians Lloyd Shapley and Jon Folkman, but was first published by the economist Ross M. Starr.

The lemma may be intuitively understood as saying that, if the number of summed sets exceeds the dimension of the vector space, then their Minkowski sum is approximately convex.

Related results provide more refined statements about how close the approximation is. For example, the Shapley–Folkman theorem provides an upper bound on the distance between any point in the Minkowski sum and its convex hull. This upper bound is sharpened by the Shapley–Folkman–Starr theorem (alternatively, Starr's corollary).

The Shapley–Folkman lemma has applications in economics, optimization and probability theory. In economics, it can be used to extend results proved for convex preferences to non-convex preferences. In optimization theory, it can be used to explain the successful solution of minimization problems that are sums of many functions. In probability, it can be used to prove a law of large numbers for random sets.

## Introductory example

A set is convex if every line segment joining two of its points is a subset in the set: For example, the solid disk $\bullet$ is a convex set but the circle $\circ$ is not, because the line segment joining two distinct points $\oslash$ is not a subset of the circle.

The convex hull of a set Q is the smallest convex set that contains Q. This distance is zero if and only if the sum is convex.

Minkowski addition is the addition of the set members. For example, adding the set consisting of the integers zero and one to itself yields the set consisting of zero, one, and two$$\{0, 1\} + \{0, 1\} = \{0 + 0, 0 + 1, 1 + 0, 1 + 1\} = \{0, 1, 2\}.$$

The subset of the integers {0, 1, 2} is contained in the interval of real numbers [0, 2], which is convex. The Shapley–Folkman lemma implies that every point in [0, 2] is the sum of an integer from {0, 1} and a real number from [0, 1].

The distance between the convex interval [0, 2] and the non-convex set {0, 1, 2} equals one-half

1/2 = \|1 − 1/2\| = \|0 − 1/2\| = \|2 − 3/2\| = \|1 − 3/2\|.

However, the distance between the average Minkowski sum

1/2 ( {0, 1} + {0, 1} ) = {0, 1/2, 1}

and its convex hull [0, 1] is only 1/4, which is half the distance (1/2) between its summand {0, 1} and [0, 1]. As more sets are added together, the average of their sum "fills out" its convex hull: The maximum distance between the average and its convex hull approaches zero as the average includes more summands.

## Preliminaries

The Shapley–Folkman lemma depends upon the following definitions and results from convex geometry.

### Real vector spaces

A real vector space of two dimensions can be given a Cartesian coordinate system in which every point is identified by an ordered pair of real numbers, called "coordinates", which are conventionally denoted by x and y. Two points in the Cartesian plane can be added coordinate-wise

(x1, y1) + (x2, y2) = (x1+x2, y1+y2);

further, a point can be multiplied by each real number λ coordinate-wise

λ (x, y) = (λx, λy).

More generally, any real vector space of (finite) dimension  $D$ can be viewed as the set of all $D$-tuples of  $D$ real numbers { (v1, v2, . . . , vD) } on which two operations are defined: vector addition and multiplication by a real number. For finite-dimensional vector spaces, the operations of vector addition and real-number multiplication can each be defined coordinate-wise, following the example of the Cartesian plane.

### Convex sets

In a real vector space, a non-empty set Q is defined to be convex if, for each pair of its points, every point on the line segment that joins them is still in Q. For example, a solid disk $\bullet$ is convex but a circle $\circ$ is not, because it does not contain a line segment joining its points $\oslash$; the non-convex set of three integers {0, 1, 2} is contained in the interval [0, 2], which is convex. For example, a solid cube is convex; however, anything that is hollow or dented, for example, a crescent shape, is non-convex. The empty set is convex, either by definition or vacuously, depending on the author.

More formally, a set Q is convex if, for all points v0 and v1 in Q and for every real number λ in the unit interval [0,1], the point

(1 − λ) v0 + λv1

is a member of Q.

By mathematical induction, a set Q is convex if and only if every convex combination of members of Q also belongs to Q. By definition, a convex combination of an indexed subset {v0, v1, . . . , vD} of a vector space is any weighted average λ0v0 + λ1v1 + . . . + λDvD, for some indexed set of non-negative real numbers {λd} satisfying the equation λ0 + λ1 + . . .  + λD = 1.

The definition of a convex set implies that the intersection of two convex sets is a convex set. More generally, the intersection of a family of convex sets is a convex set. In particular, the intersection of two disjoint sets is the empty set, which is convex.

### Convex hull

For every subset Q of a real vector space, its convex hull Conv(Q) is the minimal convex set that contains Q. Thus Conv(Q) is the intersection of all the convex sets that cover Q. The convex hull of a set can be equivalently defined to be the set of all convex combinations of points in Q. For example, the convex hull of the set of integers {0,1} is the closed interval of real numbers [0,1], which contains the integer end-points. The convex hull of the unit circle is the closed unit disk, which contains the unit circle.

### Minkowski addition

In any vector space (or algebraic structure with addition), $X$, the Minkowski sum of two non-empty sets $A, B\subseteq X$ is defined to be the element-wise operation $A+B := \{x+y\mid x\in A,~y\in B\}.$ (See also.) For example

$\{0,1\}+\{0,1\}=\{0+0,0+1,1+0,1+1\}=\{0,1,2\}$

This operation is clearly commutative and associative on the collection of non-empty sets. All such operations extend in a well-defined manner to recursive forms $\sum_{n=1}^{N}Q_{n}=Q_{1}+Q_{2}+\ldots+Q_{N}.$ By the principle of induction it is easy to see that

$\sum_{n=1}^{N}Q_{n}=\{\sum_{n=1}^{N}q_{n}\mid q_{n}\in Q_{n},~1\leq n\leq N\}.$

### Convex hulls of Minkowski sums

Minkowski addition behaves well with respect to taking convex hulls. Specifically, for all subsets $A,B\subseteq X$ of a real vector space, $X$, the convex hull of their Minkowski sum is the Minkowski sum of their convex hulls. That is,

$\mathrm{Conv}(A+B) = \mathrm{Conv}(A)+\mathrm{Conv}(B).$

And by induction it follows that

$\mathrm{Conv}(\sum_{n=1}^{N}Q_{n}) = \sum_{n=1}^{N}\mathrm{Conv}(Q_{n})$

for any $N\in\mathbb{N}$ and non-empty subsets $Q_{n}\subseteq X$, $1\leq n\leq N$.

## Statements of the three main results

### Notation

$D, N\in \{1, 2, 3, ...\}$ are positive integers.

$D$ is the dimension of the ambient space $\mathbb R^D$.

$Q_1, ..., Q_N$ are nonempty, bounded subsets of $\mathbb R^D$. They are also called "summands". $N$ is the number of summands.

$Q = \sum_{n=1}^N Q_n$ is the Minkowski sum of the summands.

$x$ is an arbitrary vector in $\mathrm{Conv}(Q)$.

### Shapley–Folkman lemma

Since $\mathrm{Conv}(Q) = \sum_{n=1}^{N}\mathrm{Conv}(Q_{n})$, for any $x \in \mathrm{Conv}(Q)$, there exist elements $q_{n}\in\mathrm{Conv}(Q_{n})$ such that $\sum_{n=1}^{N}q_{n}=x$. The Shapley–Folkman lemma refines this statement.

For example, every point in $[0,2]=[0,1]+[0,1]=\mathrm{Conv}(\{0,1\})+\mathrm{Conv}(\{0,1\})$ is the sum of an element in $\{0,1\}$ and an element in $[0,1]$.

Shuffling indices if necessary, this means that every point in $\mathrm{Conv}(Q)$ can be decomposed as

$x = \sum_{n=1}^{D}q_{n} + \sum_{n=D+1}^{N} q_n$

where $q_{n}\in\mathrm{Conv}(Q_{n})$ for $1\leq n\leq D$ and $q_{n}\in Q_{n}$ for $D+1\leq n\leq N$. Note that the reindexing depends on the point $x$.

The lemma may be stated succinctly as

<math>\mathrm{Conv}\left(\sum\_{n=1}^{N}Q\_{n}\right)\subseteq\bigcup\_{I\subseteq\\{1,2,\ldots N\\}:\~\|I\|=D}

\left( \sum\_{n\in I}\mathrm{Conv}(Q\_{n})+\sum\_{n\notin I}Q\_{n}\right).</math>

#### The converse of Shapley–Folkman lemma

In particular, the Shapley–Folkman lemma requires the vector space to be finite-dimensional.

### Shapley–Folkman theorem

Shapley and Folkman used their lemma to prove the following theorem, which quantifies the difference between $Q$ and $\mathrm{Conv}(Q)$ using squared Euclidean distance.

For any nonempty subset $S \subseteq \R^D$ and any point $x \in \R^D,$ define their squared Euclidean distance to be the infimum $d^2(x, S) = \inf_{y\in S}\|x-y\|^2.$ And more generally, for any two nonempty subsets $S, S' \subseteq \R^D,$ define $d^2(S, S') = \inf_{x\in S, y\in S'}\|x-y\|^2.$

Note that $d^2(x, S) = (\inf_{y\in S}\|x-y\|)^2,$ so we can simply write $d^2(x, S) = d(x, S)^2,$ where $d(x, S) = \inf_{y\in S}\|x-y\|.$ Similarly, $d^2(S, S') = d(S, S')^2.$

For example, $d^2([0, 2], \{0,1,2\}) = 1/4 = d([0, 2], \{0,1,2\})^2.$

The squared Euclidean distance is a measure of how "close" two sets are. In particular, if two sets are compact, then their squared Euclidean distance is zero if and only if they are equal. Thus, we may quantify how close to convexity $Q$ is by upper-bounding $d^2( \mathrm{Conv}(Q), Q).$

For any bounded subset $S \subset \R^D,$ define its circumradius $rad(S)$ to be the infimum of the radius of all balls containing it (as shown in the diagram). More formally, $rad(S) \equiv \inf_{x \in R^N} \sup_{y \in S}\|x-y\|$ Now we can state

where we use the notation $\sum_{\max D}$ to mean "the sum of the $D$ largest terms".

Note that this upper bound depends on the dimension of ambient space and the shapes of the summands, but not on the number of summands.

### Shapley–Folkman–Starr theorem

Define the inner radius $r(S)$ of a bounded subset $S \subset \R^D$ to be the infimum of $r$ such that, for any $x\in \mathrm{Conv}(S)$, there exists a ball $B$ of radius $r$ such that $x\in \mathrm{Conv}(S \cap B)$.

For example, let $B' \subset B \subset \R^D$ be two nested balls, then the circumradius of $B\setminus B'$ is the radius of $B$, but its inner radius is the radius of $B'$.

Since $r(S) \leq rad(S)$ for any bounded subset $S \subset \R^D$, the following theorem is a refinement:

In particular, if we have an infinite sequence $(Q_n)_{n=1, 2, ...}$ of nonempty, bounded subsets of $\mathbb R^D$, and if there exists some $r_0 \geq 0$ such that the inner radius of each $Q_n$ is upper-bounded by $r_0$, then $d^2\left( \mathrm{Conv}\left(\frac 1N \sum_{n=1}^N Q_n \right), \frac 1N \sum_{n=1}^N Q_n \right)
\leq
\frac{D r_0^2}{N}.$This can be interpreted as stating that, as long as we have an upper bound on the inner radii, performing "Minkowski-averaging" would get us closer and closer to a convex set.

### Other proofs of the results

There have been many proofs of these results, from the original, to the later Arrow and Hahn, Cassels, Schneider, etc. An abstract and elegant proof by Ekeland has been extended by Artstein. Different proofs have also appeared in unpublished papers. An elementary proof of the Shapley–Folkman lemma can be found in the book by Bertsekas, together with applications in estimating the duality gap in separable optimization problems and zero-sum games.

Usual proofs of these results are nonconstructive: they establish only the existence of the representation, but do not provide an algorithm for computing the representation. In 1981, Starr published an iterative algorithm for a less sharp version of the Shapley–Folkman–Starr theorem.

### A proof of the results

The following proof of Shapley–Folkman lemma is from. The proof idea is to lift the representation of $x$ from $\R^{D}$to $\R^{D+N}$, use Carathéodory's theorem for conic hulls, then drop back to $\R^D$.

The following "probabilistic" proof of Shapley–Folkman–Starr theorem is from.

We can interpret $\mathrm{Conv}(S)$ in probabilistic terms: $\forall x \in \mathrm{Conv}(S)$, since $x = \sum w_n q_n$ for some $q_n \in S$, we can define a random vector $X$, finitely supported in $S$, such that $Pr(X=q_n) = w_n$, and $x = \mathbb E[X]$.

Then, it is natural to consider the "variance" of a set $S$ as$Var(S) := \sup_{x \in \mathrm{Conv}(S)} \inf_{\mathbb E[X] = x, X\text{ is finitely supported in }S} Var[X]$With that, $d(S, \mathrm{Conv}(S))^2 \leq Var(S) \leq r(S) \leq rad(S)$.

## History

The lemma of Lloyd Shapley and Jon Folkman was first published by the economist Ross M. Starr, who was investigating the existence of economic equilibria while studying with Kenneth Arrow. In his paper, Starr studied a convexified economy, in which non-convex sets were replaced by their convex hulls; Starr proved that the convexified economy has equilibria that are closely approximated by "quasi-equilibria" of the original economy; moreover, he proved that every quasi-equilibrium has many of the optimal properties of true equilibria, which are proved to exist for convex economies.

Following Starr's 1969 paper, the Shapley–Folkman–Starr results have been widely used to show that central results of (convex) economic theory are good approximations to large economies with non-convexities; for example, quasi-equilibria closely approximate equilibria of a convexified economy. "The derivation of these results in general form has been one of the major achievements of postwar economic theory", wrote Roger Guesnerie.

The topic of non-convex sets in economics has been studied by many Nobel laureates: Shapley himself (2012), Arrow (1972), Robert Aumann (2005), Gérard Debreu (1983), Tjalling Koopmans (1975), Paul Krugman (2008), and Paul Samuelson (1970); the complementary topic of convex sets in economics has been emphasized by these laureates, along with Leonid Hurwicz, Leonid Kantorovich (1975), and Robert Solow (1987).

## Applications

The Shapley–Folkman lemma enables researchers to extend results for Minkowski sums of convex sets to sums of general sets, which need not be convex. Such sums of sets arise in economics, in mathematical optimization, and in probability theory; in each of these three mathematical sciences, non-convexity is an important feature of applications.

### Economics

In economics, a consumer's preferences are defined over all "baskets" of goods. Each basket is represented as a non-negative vector, whose coordinates represent the quantities of the goods. On this set of baskets, an indifference curve is defined for each consumer; a consumer's indifference curve contains all the baskets of commodities that the consumer regards as equivalent: That is, for every pair of baskets on the same indifference curve, the consumer does not prefer one basket over another. Through each basket of commodities passes one indifference curve. A consumer's preference set (relative to an indifference curve) is the union of the indifference curve and all the commodity baskets that the consumer prefers over the indifference curve. A consumer's preferences are convex if all such preference sets are convex.

An optimal basket of goods occurs where the budget-line supports a consumer's preference set, as shown in the diagram. This means that an optimal basket is on the highest possible indifference curve given the budget-line, which is defined in terms of a price vector and the consumer's income (endowment vector). Thus, the set of optimal baskets is a function of the prices, and this function is called the consumer's demand. If the preference set is convex, then at every price the consumer's demand is a convex set, for example, a unique optimal basket or a line-segment of baskets.

#### Non-convex preferences

However, if a preference set is non-convex, then some prices determine a budget-line that supports two separate optimal-baskets. For example, we can imagine that, for zoos, a lion costs as much as an eagle, and further that a zoo's budget suffices for one eagle or one lion. We can suppose also that a zoo-keeper views either animal as equally valuable. In this case, the zoo would purchase either one lion or one eagle. Of course, a contemporary zoo-keeper does not want to purchase half of an eagle and half of a lion (or a griffin)! Thus, the zoo-keeper's preferences are non-convex: The zoo-keeper prefers having either animal to having any strictly convex combination of both.

When the consumer's preference set is non-convex, then (for some prices) the consumer's demand is not connected; a disconnected demand implies some discontinuous behavior by the consumer, as discussed by Harold Hotelling:

> If indifference curves for purchases be thought of as possessing a wavy character, convex to the origin in some regions and concave in others, we are forced to the conclusion that it is only the portions convex to the origin that can be regarded as possessing any importance, since the others are essentially unobservable. They can be detected only by the discontinuities that may occur in demand with variation in price-ratios, leading to an abrupt jumping of a point of tangency across a chasm when the straight line is rotated. But, while such discontinuities may reveal the existence of chasms, they can never measure their depth. The concave portions of the indifference curves and their many-dimensional generalizations, if they exist, must forever remain in unmeasurable obscurity.

The difficulties of studying non-convex preferences were emphasized by Herman Wold and again by Paul Samuelson, who wrote that non-convexities are "shrouded in eternal darkness ...", according to Diewert.

Nonetheless, non-convex preferences were illuminated from 1959 to 1961 by a sequence of papers in The Journal of Political Economy (JPE). The main contributors were Farrell, Bator, Koopmans, and Rothenberg. In particular, Rothenberg's paper discussed the approximate convexity of sums of non-convex sets. These JPE-papers stimulated a paper by Lloyd Shapley and Martin Shubik, which considered convexified consumer-preferences and introduced the concept of an "approximate equilibrium". The JPE-papers and the Shapley–Shubik paper influenced another notion of "quasi-equilibria", due to Robert Aumann.

#### Starr's 1969 paper and contemporary economics

Previous publications on non-convexity and economics were collected in an annotated bibliography by Kenneth Arrow. He gave the bibliography to Starr, who was then an undergraduate enrolled in Arrow's (graduate) advanced mathematical-economics course. In his term-paper, Starr studied the general equilibria of an artificial economy in which non-convex preferences were replaced by their convex hulls. In the convexified economy, at each price, the aggregate demand was the sum of convex hulls of the consumers' demands. Starr's ideas interested the mathematicians Lloyd Shapley and Jon Folkman, who proved their eponymous lemma and theorem in "private correspondence", which was reported by Starr's published paper of 1969.

In his 1969 publication, Starr applied the Shapley–Folkman–Starr theorem. Starr proved that the "convexified" economy has general equilibria that can be closely approximated by "quasi-equilibria" of the original economy, when the number of agents exceeds the dimension of the goods: Concretely, Starr proved that there exists at least one quasi-equilibrium of prices popt with the following properties:

- For each quasi-equilibrium's prices popt, all consumers can choose optimal baskets (maximally preferred and meeting their budget constraints).
- At quasi-equilibrium prices popt in the convexified economy, every good's market is in equilibrium: Its supply equals its demand.
- For each quasi-equilibrium, the prices "nearly clear" the markets for the original economy: an upper bound on the distance between the set of equilibria of the "convexified" economy and the set of quasi-equilibria of the original economy followed from Starr's corollary to the Shapley–Folkman theorem.

Starr established that

> "in the aggregate, the discrepancy between an allocation in the fictitious economy generated by [taking the convex hulls of all of the consumption and production sets] and some allocation in the real economy is bounded in a way that is independent of the number of economic agents. Therefore, the average agent experiences a deviation from intended actions that vanishes in significance as the number of agents goes to infinity".

Following Starr's 1969 paper, the Shapley–Folkman–Starr results have been widely used in economic theory. Roger Guesnerie summarized their economic implications: "Some key results obtained under the convexity assumption remain (approximately) relevant in circumstances where convexity fails. For example, in economies with a large consumption side, preference nonconvexities do not destroy the standard results". "The derivation of these results in general form has been one of the major achievements of postwar economic theory", wrote Guesnerie. The topic of non-convex sets in economics has been studied by many Nobel laureates: Arrow (1972), Robert Aumann (2005), Gérard Debreu (1983), Tjalling Koopmans (1975), Paul Krugman (2008), and Paul Samuelson (1970); the complementary topic of convex sets in economics has been emphasized by these laureates, along with Leonid Hurwicz, Leonid Kantorovich (1975), and Robert Solow (1987). The Shapley–Folkman–Starr results have been featured in the economics literature: in microeconomics, in general-equilibrium theory,[^1] in public economics (including market failures), as well as in game theory, in mathematical economics, and in applied mathematics (for economists). The Shapley–Folkman–Starr results have also influenced economics research using measure and integration theory.

### Mathematical optimization

The Shapley–Folkman lemma has been used to explain why large minimization problems with non-convexities can be nearly solved (with iterative methods whose convergence proofs are stated for only convex problems). The Shapley–Folkman lemma has encouraged the use of methods of convex minimization on other applications with sums of many functions.

#### Preliminaries of optimization theory

Nonlinear optimization relies on the following definitions for functions:

- The graph of a function f is the set of the pairs of arguments x and function evaluations f(x)

Graph(f) = { (x, f(x) ) }

- The epigraph of a real-valued function f is the set of points above the graph

Epi(f) = { (x, u) : f(x) ≤ u }.

- A real-valued function is defined to be a convex function if its epigraph is a convex set.

For example, the quadratic function f(x) = x2 is convex, as is the absolute value function g(x) = \|x\|. However, the sine function (pictured) is non-convex on the interval (0, π).

#### Additive optimization problems

In many optimization problems, the objective function f is separable: that is, f is the sum of many summand-functions, each of which has its own argument:

f(x) = f( (x1, ..., x$N$) ) = Σ fn(xn).

For example, problems of linear optimization are separable. Given a separable problem with an optimal solution, we fix an optimal solution

xmin = (x1, ..., x$N$)min

with the minimum value f(xmin). For this separable problem, we also consider an optimal solution (xmin, f(xmin) ) to the "convexified problem", where convex hulls are taken of the graphs of the summand functions. Such an optimal solution is the limit of a sequence of points in the convexified problem

(xj, f(xj) ) ∈  Σ Conv (Graph( fn ) ).

Of course, the given optimal-point is a sum of points in the graphs of the original summands and of a small number of convexified summands, by the Shapley–Folkman lemma.

This analysis was published by Ivar Ekeland in 1974 to explain the apparent convexity of separable problems with many summands, despite the non-convexity of the summand problems. In 1973, the young mathematician Claude Lemaréchal was surprised by his success with convex minimization methods on problems that were known to be non-convex; for minimizing nonlinear problems, a solution of the dual problem need not provide useful information for solving the primal problem, unless the primal problem be convex and satisfy a constraint qualification. Lemaréchal's problem was additively separable, and each summand function was non-convex; nonetheless, a solution to the dual problem provided a close approximation to the primal problem's optimal value. Ekeland's analysis explained the success of methods of convex minimization on large and separable problems, despite the non-convexities of the summand functions. Ekeland and later authors argued that additive separability produced an approximately convex aggregate problem, even though the summand functions were non-convex. The crucial step in these publications is the use of the Shapley–Folkman lemma.

`The Shapley–Folkman lemma has encouraged the use of methods of convex minimization on other applications with sums of many functions.`

### Probability and measure theory

Convex sets are often studied with probability theory. Each point in the convex hull of a (non-empty) subset Q of a finite-dimensional space is the expected value of a simple random vector that takes its values in Q, as a consequence of Carathéodory's lemma. Thus, for a non-empty set Q, the collection of the expected values of the simple, Q-valued random vectors equals Q's convex hull; this equality implies that the Shapley–Folkman–Starr results are useful in probability theory. In the other direction, probability theory provides tools to examine convex sets generally and the Shapley–Folkman–Starr results specifically. The Shapley–Folkman–Starr results have been widely used in the probabilistic theory of random sets, for example, to prove a law of large numbers, a central limit theorem, and a large-deviations principle. These proofs of probabilistic limit theorems used the Shapley–Folkman–Starr results to avoid the assumption that all the random sets be convex.

A probability measure is a finite measure, and the Shapley–Folkman lemma has applications in non-probabilistic measure theory, such as the theories of volume and of vector measures. The Shapley–Folkman lemma enables a refinement of the Brunn–Minkowski inequality, which bounds the volume of sums in terms of the volumes of their summand-sets. The volume of a set is defined in terms of the Lebesgue measure, which is defined on subsets of Euclidean space. In advanced measure-theory, the Shapley–Folkman lemma has been used to prove Lyapunov's theorem, which states that the range of a vector measure is convex. Here, the traditional term "range" (alternatively, "image") is the set of values produced by the function. A vector measure is a vector-valued generalization of a measure; for example, if p1 and p2 are probability measures defined on the same measurable space, then the product function p1 p2 is a vector measure, where p1 p2 is defined for every event ω by

(p1 p2)(ω)=(p1(ω), p2(ω)).

Lyapunov's theorem has been used in economics, in ("bang-bang") control theory, and in statistical theory. Lyapunov's theorem has been called a continuous counterpart of the Shapley–Folkman lemma, which has itself been called a discrete analogue of Lyapunov's theorem.

[^1]:

Sums of sets of vectors are nearly convex