text:
Star Trek: The Next Generation – The Price (Review)
This January and February, we'll be finishing up our look at the second season of Star Trek: The Next Generation and moving on to the third year of the show, both recently and lovingly remastered for high definition. Check back daily for the latest review.
Well, the streak had to end some time. After seven episodes ranging from "flawed but still interesting" to "pretty great", the third season of Star Trek: The Next Generation hits a bit of a snag. The Price is the weakest episode of the show's third season to this point, and confirmation that the writers really have no idea how to write for Deanna Troi. It's still the best episode to focus on the ship's half-Betazoid counsellor, but being better than Haven or The Child is hardly an accomplishment for the ages.
The Next Generation had a lot of trouble with its female characters. In the first season, casting Tasha Yar as the ship's security chief and tactical officer was a nice way of subverting gender conventions, but all the show ever did with her was have her get all hot and bothered about sexy men. The fact that the first two episodes with any focus on Yar were Code of Honour and The Naked Now doesn't help matters, but Yar was hardly professional in Justice. And that's without getting into how tone deaf Yar's characterisation was when paired with the show's decision to make her a survivor of sexual assault.
However, once Denise Crosby departed the show in Skin of Evil, the only two female members of the ship's senior staff were the ship's therapist and the medical officer. It's hard to imagine more stereotypical gender roles on the Enterprise. Again, this wouldn't be a problem in and of itself; it's the combination of all these different factors. While Crusher (and even Pulaski) have headlined episodes that treat them as characters in their own right, and play particularly well in the ensemble, Troi is a character who is defined as stereotypically feminine.
Tabling the talks…
Her first character-centric episode concerned her arranged marriage and domineering mother; Troi was so incredibly passive in Haven that the episode's plot seemed to happen around her. Then, in The Child, Troi was used by an alien intelligence as a walking incubation centre. The Price at least escalates Troi to the role of romantic lead, but the whole episode seems to be built around the fact that Troi's defining character attribute is that she's "the girl" on the bridge, in the most sexist manner possible.
The Price opens like any terrible romantic movie, with Troi returning home unappreciated after a hard day's work. She discovers that her domineering mother has been sending her emails and that she can't have any chocolate. She's just about to try to relax when her thoughtlessly jovial emotionally inattentive boss (who happens to be British) imposes on her to come to a work function. Patrick Stewart seems to realise that he's playing a quirky romantic supporting character, so he plays up the scene beautifully, instructing Troi to show up. "Just throw on any old thing!" he insists, playfully, which might be the least Picard moment ever.
Surprisingly, the Ferengi are the least of the episode's problems…
And all this is before we meet Ral. Who is the first proper indication of just how terrible Troi's taste in men happens to be. In his introductory scene, he is introduced with a beautiful woman draped over his arm. On meeting Troi, he promptly sends his girlfriend packing, which is always a good sign in a romantic partner. Ral goes into full on creep mode, and it's incredible how Troi seems to respond to Ral's aggressive flirtation and terrible lines. The Price's idea of romance is cringe-inducing. "Am I moving too fast for you?" Ral asks at one point. "No," Troi responds, "I'm moving too fast for me."
There are lots of problems with this basic set-up. Ral is a pretty unlikeable character. He's pretty much the archetypal eighties capitalist douchebag, which means that he's really just the Ferengi with a more handsome exterior. His seduction technique is basically to walk into Troi's office, insult her a bit, grope her, mess with her hair and then invite her to dinner. He seems to be The Next Generation's version of the classic romantic "bad boy" archetype, just sterilised and safely desexualised for the sexless Next Generation.
Through the wormhole…
It's very weird that Troi seems to respond to him. There is interesting drama to be written in having a smart character make dumb romantic choices. Love is something that blinds people, and makes it possible to catch them off-guard. However, as with so many of Troi's other romantic choices, her attraction to Ral makes absolutely no sense. It's impossible to see why she would be remotely drawn to him in the first place, save that he looks sort of handsome. For a character whose place on the ship relies quite heavily on her advanced emotional intelligence, that is not good characterisation at all.
There is nothing to like about Ral, which makes the whole plot feel a little pointless. In the final act, we're supposed to believe that Troi's unilateral decision to "out" Ral is – in some small measure – for his own good. "I'm very grateful for what you did, in a way," Ral tells her in the final scene. "It's made me take a hard look at who I am. I don't like what I see." The implication is that Troi saw something in him that was worth trying to save. Unfortunately, due to the scripting and performances, it plays more like a fit of pique from Troi, as if she's decided to lash out at Ral for sabotaging Riker.
From Patrick Stewart's delivery, I suspect that is not Picard's first champagne flute…
After all, Ral is presented pretty consistently and thoroughly as an amoral capitalist. What rationalisations he might provide to Troi, the audience gets to see Ral trying to use his girlfriend as emotional leverage on Riker. "It's just that, well, she's an extraordinary woman," Ral boasts. "Brilliant. Lovely. Very passionate. And she could have been yours, Will, but you just didn't do enough to keep her. And now, well, I'm here, and I'm going to take her too." Because there's nothing sexier than a man who treats a woman like property, am I right?
(It's to Riker's credit that he doesn't rise to the bait and allows Deanna her own choices. "If you can bring happiness into Deanna's life, nothing would please me more," he concedes. "Deanna is just the woman to bring some meaning to your sorry existence, if you're smart enough to take it. I doubt that you are." Still, the entire sequence exists just to reinforce the idea that Ral is a sexist selfish sociopath with no regard for anybody but himself. If only there were a skilled psychiatrist on board who could read the signs.)
Good talk.
Then there's Star Trek's first sex scene. It generated some small measure of controversy when it was first suggested, but Michael Piller has proudly boasted about how the final cut managed to avoid offending anybody. Given how poorly The Next Generation does sensuous and sexual, it's no surprise that the entire sequence is horrifying. It's very eighties, with terrible dialogue and lots of very tasteful night clothes and absolutely zero passion. Marina Sirtis is not the strongest member of the Next Generation ensemble, and she has absolutely no chemistry with Matt McCoy. There's none of the heat of any of Shatner's make-out scenes.
To be fair to Ral, he suffers because he's written as a generic amoral capitalist in Roddenberry's socialist utopia future. This is the show that thought the Ferengi might be a credible threat to the Enterprise. So, naturally, a man who uses his gifts for the purposes of commercial trade is the worst sort of villain in the world. In a way, The Price feels like an artefact from the first or second season, where any value system that does not match that of the Federation is treated as inherently hostile. "You know, you're really not such a bad sort, Ral," Riker offers, "except you don't have any values, beyond the value of today's bid, that is."
Oh, I'm shaking in my little space boots…
When Troi discovers that Ral is part Betazoid, and thus empathic, she is appalled. "But you're reading their emotional states, their inner selves, and then using that to manipulate them," Troi protests. Ral responds, "Well, people have been doing that for thousands of years, just by listening carefully, by watching body language. I just happen to be better at it. You do it." That's a very valid argument, particularly when Picard compares negotiations to high-stakes poker.
Surely the same principle applies to Troi's empathic abilities at the poker table? Or Geordi's VISOR, which can tell if people are lying? Or Data's perfect poker face and mathematical abilities? Troi glosses over that, and jumps straight to the high moral ground. "I do it to help my crew, not outmanoeuvre them. And I don't hide that I'm an empath." Ral cuts across her, "Oh, so you announce it to every alien culture you encounter? Or do you use it to give your side an advantage. Do you tell the Romulan that's about to attack that you sense that he may be bluffing? Or do you just tell it to your Captain?"
"Yeah, Deep Space Nine's is way more impressive…"
"That's different," Troi responds. "That's a matter of protection." In other words, her actions are justified because they are in service of Starfleet's agenda. Which is a disturbingly partisan way of looking at the universe. "Yes, protection," Ral repeats. "Your protection, your Captain, your crew, your edge. Yes. Now it's a matter of life and death when you take the advantage. Me, I deal in property. Exchanges. No body gets hurt. So you tell me, which one of us would you say has more of a problem with ethics?"
The problem is that Ral isn't inherently wrong here. His position is valid. He uses his gift in a way that is not too different from the way that Troi uses her gift. Who is to say that financial and economic expansion is any more or any less valid than the Federation's approach to galactic peace? It is willing to pay for use of the wormhole, so it's not as if the Federation has gone beyond such bartering and negotiation.
Following the empath of least resistance…
Troi's own mother is a respected Federation diplomat, and Manhunt features her using her powers to read the minds of some delegates without their permission. She saves the conference in doing so, but doesn't that give her an even more unfair advantage than that held by Ral? The Price can't really make that argument, so it instead suggests that Ral is an amoral sleazy selfish manipulative self-righteous sociopath. There's a sense that he's being portrayed as a jerk primary because he adheres to a value system slightly different to that of the Federation. It's the same problem the show faced with the Ferengi in the first season.
It's a shame, because The Price does have an interesting plot outside of Troi's awful romance. Indeed, two of the core concepts of The Price would go on to drive the next two Star Trek spin-offs. Star Trek: Deep Space Nine would be built around the discovery of the first stable wormhole known to exist, linking the Alpha and Gamma Quadrants. Star Trek: Voyager would feature a starship lost in the Delta Quadrant. It even produced a sequel to The Price.
What does Riker bring to the table?
At the heart of The Price are the negotiations over the Barzan wormhole. Barzan itself is presented as a backwater planet with no real political influence or galactic importance. "And as you all know, the environment on my planet is completely inhospitable to most other life forms," Bhavani explains. "So I'd like to express my appreciation to you, Captain Picard, for hosting these negotiations."
However, Barzan's ambitions extend beyond this. "The Barzan has been a society dependent on others for generations. We want that to end. The appearance of this stable wormhole in our space provides us with our first true natural resource. We have neither the experience nor the technology to exploit it. But you do." It's a nice metaphor for the exploitation of natural resources in the third world by more "developed" countries – oil or gas or forestry.
The chair man…
Given the dialogue that existed in the late eighties about third world debt and the "oil glut", this is quite a potent metaphor. The Federation is frequently used as a stand-in for the United States, so this would be the perfect opportunity to explore the relationship between the Federation and less advanced species who have materials that the Federation might require. How does the Federation determine a "fair price"? Can the organisation be as altruistic as it claims?
Unfortunately, The Price isn't interested in this rather weighty subtext. Instead, it spends most of its forty-five minutes as a romance with a comedy subplot involving the Ferengi. The Price is interesting in this regard. In one sense, The Price represents the best use of the Ferengi to date. We're well past the point where the Ferengi can be used as credible villains, and it makes sense that a bunch of ruthless capitalists could prove a threat to the Federation around the negotiation table. The Price feels like a much better fit for the Ferengi than Peak Performance.
I wonder if Picard feels the same dread that the audience do whenever they see that ship…
On the other hand, the comedy involving the Ferengi doesn't quite work. They are portrayed as a little too comedic, a little too buffoonish. "Idiots!" Geordi protests, and it's hard to disagree. The Price gives us a couple of gags that work – Goss showing up and demanding a chair, for example – but quite a few more that don't. Most obviously, there's a sense that the Ferengi are miscast. The actors don't really seem to have the necessary timing.
In particular, Scott Thomson seems to struggle to articulate through his prosthetic teeth, which kills lines that might have worked otherwise like "I have no wish to kill anyone – a short term crippling will suffice." This isn't the biggest problem in the world. After all, the cast of The Next Generation used to joke that Michael Dorn could never nail a first take through Worf's prosthetic teeth, and he was wearing the same teeth for seven years. We can forgive a guest actor for messing up a few line readings under those conditions; it's just one more disappointment in an episode that never quite works.
Kissing romance goodbye…
The Price is the one of the weakest episodes of the third season, but it stands in sharp contrast to the shows constructed around it. By this point, the first season had given us The Naked Now, Code of Honour, The Last Outpost, Lonely Among Us and Justice. The second season didn't fare too much better, producing The Child, The Outrageous Okona and Unnatural Selection in its first eight episodes. If The Price is the only true misfire at this point in the thirds season, something must be going right.
Read our reviews of the third season of Star Trek: The Next Generation:
Supplemental: (DC Comics, 1989) #1-2 – Return to Raimon/Murder, Most Foul
The Ensigns of Command
Supplemental: The Ensigns of Command by Melinda Snodgrass
The Bonding
Supplemental: Star Trek (DC Comics, 1989) #19 – Once a Hero…
Supplemental: (DC Comics, 1989) #19 – The Lesson
Booby Trap
Supplemental: The Romulan Way by Diane Duane and Peter Morwood
The Vengeance Factor
Supplemental: The Sky's the Limit – Suicide Note by Geoff Trowbridge
The High Ground
Déjà Q
Supplemental: (DC Comics) Annual #1 – The Gift
Supplemental: I, Q by John DeLancie and Peter David
A Matter of Perspective
Yesterday's Enterprise
Supplemental: The Lost Era – Well of Souls by Ilsa J. Bick
Supplemental: The Last Generation
Supplemental: Q-Squared by Peter David
Supplemental: Phase II (1978) – Kitumba, Parts I & II
Captain's Holiday
Hollow Pursuits
Supplemental: Sarek by A.C. Crispin
Ménage à Troi
Supplemental: Imzadi by Peter David
Transfigurations
Supplemental: Star Trek/X-Men: Star TreX
The Best of Both Worlds, Part I
Supplemental: (DC Comics, 1989) #47-50 – The Worst of Both Worlds
Supplemental: Vendetta by Peter David
Filed under: The Next Generation | Tagged: Captain Picard, christmas, Deanna, Deanna Troi, Denise Crosby, games, GOSS, jean-luc picard, Next Generation, patrick stewart, picard, Price, Romulan, science fiction, star trek, Star Trek Next Generation, StarTrek, Tasha Yar, Troi, Video game, William Riker |
« Star Trek: The Lost Era – Serpents Among the Ruins by David R. George III (Review) Jameson Cult Film Club Screening of Jaws! With Richard Dreyfuss in Attendance! Tuesday 18th February! »
mgpogue, on February 5, 2014 at 7:03 pm said:
Reblogged this on The TV Media Junkie ReBlog Blog.
Zeno, on February 5, 2014 at 10:23 pm said:
This is off topic problem I need some help for. My ebook copy of the Final Reflection might be missing some passages. Is there anything going on between Vreen going to bed and the description of the Romulan being tortured in chapter 3? There is another odd gap with Vreen walking with the commander to the turbo lift and then talking to the new roommate. That can't be right. Are there missing scenes?
Darren, on February 7, 2014 at 12:11 am said:
Sorry about delay. Had to go digging.
On the first one, my ebook copy has the lines:
He was instantly asleep.
Security had a Rom in the cube.
There's definitely a gap there, but I don't think there's anything missing. It's just a scene transition.
However, my ebook does have a weird line format on the lift scene:
he saw one of the Bridge crew flash him the spread fingers
course . . . of the Roms around it.
of the Captain's Star, and then another, and another.
That second line shouldn't be there, but the scene reads fluidly without it. I suspect my ebook edition may have needed more rigorous editing. (Although it is quite an old copy.)
Zeno, on February 7, 2014 at 3:10 am said:
So there is a gap? My copy is a pdf file which was made from a .rtf of the original book.
Darren, on February 12, 2014 at 1:05 pm said:
Well, there's definitely an error in my copy of the second scene. I think the first is just a scene transition.
Ross Nolan, on February 5, 2014 at 10:42 pm said:
'It's very eighties, with terrible dialogue and lots of very tasteful night clothes and absolutely zero passion.'
The eighties had zero passion? Obviously someone never watched Dallas – Larry Hagman and Linda Grey could set fire to the scenery from ten paces away. 😉
I do find it depressing going back over the show to be reminded how sexist TNG was in the early years, even by the standards of the decade it was created in.
Yep. And it gets a little better, but it's never entirely cured. The last season features Sub Rosa, an episode about Beverly Crusher's sexy Scottish ghost, in which she masturbates with a candle. (As tastefully as one can do so on television in the mid-nineties.)
Robert Bateman, on September 30, 2015 at 3:03 am said:
In the first sentence in the paragraph above the photograph titled "Oh, I'm shaking in my little space boots…" the sentence needs to be changed to "To be fair to Ral, he suffers because he's written as a generic amoral capitalist in Roddenberry's socialist utopia future."
Darren, on October 1, 2015 at 12:17 pm said:
Cheers, corrected.


text:
Q: Why is one of the two strings below considered as a JSON array and the other one as a string? Python I am trying to make a PUT request and the API endpoint requires the payload to be structured as below:
payload_test = "[{\"id\": 1, \"channel_id\": 1, \"entity_type\": \"product\", \"entity_id\": 123, \"file_name\": \"custom-product-1.html\"}]"

I have imported values from a CSV file to create a similar string response using pandas and some string manipulation. This is my output when I print my string:
my_string = "[{\"id\": 1, \"channel_id\": 1, \"entity_type\": \"product\", \"entity_id\": 123, \"file_name\": \"product-custom-options.html\"}]"

I have created my_string to be identical to the original payload request but when I pass my_string I receive the following error: The request payload has to be a JSON array for the endpoint
The api call is succesful when I pass payload_test variable instead
This is how I created the my_string string:
The values for the id, channel_id etc are obtained from a dataframe (which is orgianally a csv file).

id
channel_id
entity_type
entity_id
file_name

1
1
product
123
custom-product-1.html

2
2
product
234
custom-product-1.html

df = pd.read_csv('xyz.csv')

for trial in df[['id', 'channel_id', 'entity_type', 'entity_id', 'file_name']].agg(pd.Series.to_json, axis='columns'):

    string1 = json.dumps(trial)

    my_string = string1[:1] + '[' + string1[1:-1] + ']"' #adding [ and ]" to the beginning and end

    my_string = my_string.replace(':', ': ')   #adding spaces after :

    my_string = my_string.replace(',', ', ')   #adding spaces after ,

    update_template(my_string, headers)

When I print the type for both the strings, it is: <class 'str'>

A: Because pd.Series.to_json returns a string, not a JSON-like object (would have been a dict in this case). From the documentation:

pandas.Series.to_json
Convert the object to a JSON string.

Calling json.dumps on a string enclose it in a pair of double quotes and mess up everything after. Try this instead:
for trial in df[['id', 'channel_id', 'entity_type', 'entity_id', 'file_name']].agg(pd.Series.to_json, axis='columns'):
    my_string = f'[{trial}]'.replace(':', ': ').replace(',', ', ')

The two replaces are not necessary and may even interfere with the fields' content.

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All of these works have been made in collaboration with dramaturge Therese Steele.
Tatge&Steele's collaboration is based on a mutual interest in exploring notions of power and identity, the politics of the subconscious and the relationship between individual and collective, through a variety of medias including performance, installation and film.
'Networking' November 2012, was commissioned by Arts Admin London. The work is a satirical intervention-based performance questioning the position of being female in the current art market.
'Silent dinner' March 2013, was an attempt to utilise the act of eating as a channel into the sensorial, whilst emphasising the gestural and guttural act of food consumption itself.
'All in Search of a Greater Hunger' April 2014, was their second work presented at Freud Museum in London. Situated in and around Miroslav Balka's DIE TRAUMDEUTUNG (the dream interpretation) 75,32m AMSL, the piece accessed Balka's installation as a site for exploring the relation between material surroundings and mental states.


Receive an overview of the history of environmental change in the Park, emphasizing human-caused change. Discuss the history of mining, water engineering, recreational development and tourism, atmospheric inputs of dust and nitrates, wildlife management, and climate change in the context of protecting and restoring natural environments within the Park. Understand how past human activities influence contemporary ecosystems and options for management of natural resources within the Park.


text:
Chelsea gain revenge on Leicester to take control of top-four fate
The 8,000 crowd saw the Blues keep a place in next season's Champions League in their own hands thanks to second-half goals from Antonio Rudiger and Jorginho.
Chelsea beat Leicester 2-1 on 18 May 2021 in front of 8,000 fans at the Stamford Bridge in London. Picture: @ChelseaFC/Twitter.
AFP | 244 days ago
LONDON, United Kingdom - Chelsea manager Thomas Tuchel thanked the return of fans to Stamford Bridge for helping side avenge their FA Cup final defeat to Leicester with a 2-1 win over the Foxes to move into third in the Premier League on Tuesday.
Victory away to Aston Villa on Sunday will guarantee Chelsea a top-four finish.
"It was an outstanding performance but the job is not done," said Tuchel.
"The fans made a huge difference. The speed in our game, the hunger, the ambition. It was a very strong performance and I'm delighted the team can present itself like this in front of our fans."
Leicester will forever have the memories of winning the FA Cup for the first time at Wembley on Saturday, but are now at risk of missing out on Champions League football on the final day of the campaign for the second consecutive season.
Brendan Rodgers's men still lead Liverpool by three points, but the Reds will go fourth on goal difference if they win at Burnley on Wednesday.
Leicester host Tottenham on the final day, while Jurgen Klopp's men welcome Crystal Palace to Anfield.
"We have to get the points on Sunday then see where we are. If we don't get in the Champions League it has still been a great season," said Rodgers.
"If we just fall short after 38 games as the team with the eighth biggest budget, it shows we fought really hard and just missed out. There are still points to play for and we'll be ready for that."
Chelsea's final league game under Frank Lampard was a 2-0 defeat away to Leicester in January that opened up a nine-point gap between the sides.
The 10-point swing since is illustrative of the impact Tuchel's arrival has had on the Londoners with their first major disappointment under the German coming at the weekend.
Roared on by the biggest crowd at Stamford Bridge for 14 months, Chelsea exploded out the blocks with Leicester barely able to get out their own half in the first 45 minutes.
However, a combination of goalkeeper Kasper Schmeichel and controversial refereeing kept the hosts at bay until the break.
Timo Werner had two goals disallowed in the first half as the German international was firstly flagged offside before his second effort was ruled out by a VAR review for handball.
In between times, Werner was also dismayed that he was denied a penalty after being caught by Youri Tielemans inside the box.
TEMPERS FLARE
With millions on the line for a place in the Champions League, that decision could have proved very costly for Chelsea.
But there was little doubt about the outcome once Rudiger bundled home a corner two minutes into the second half.
Werner finally got a VAR call to go his way to double Chelsea's lead 24 minutes from time.
Wesley Fofana's foul on Werner was initially given as a free-kick outside the box, but on review referee Mike Dean was told to point to the penalty spot.
Jorginho coolly sent Schmeichel the wrong way and for 10 minutes, Chelsea looked to be cruising.
However, Leicester set up a nervy finale when Wilfred Ndidi robbed Mateo Kovacic and teed up Iheanacho to slot home his 19th goal of the season.
The Nigerian also became the first player to score on all seven days of the week in a single Premier League season.
Leicester had a glorious chance to salvage a point and keep their top-four fate in their own hands in the final minute, but Ayoze Perez blasted over from Ricardo Pereira's cut-back.
Tempers flared in stoppage time with both sets of players, substitutes and coaches involved in an altercation after Rudiger took exception to a challenge from Pereira on Ben Chilwell.
Chelsea, though, kept their heads to see out the final few seconds and take a huge step towards another season of Champions League football.
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text:
Q: A 3D polygon with sides tangent to a sphere Imagine the quadrilateral in the picture placed on a sphere of size that allow the four sides to touch the sphere, in other words: the four sides of this 3D polygon are tangent to the sphere. The 4 points where the sides touch the sphere are on a circle. This has been presented here.
My question is: Are always the tangent points of any n side polygon on a circle?


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text:
'M' for multiracial — Dr M's one word summing up Malaysia
'Malaysians are too focused on race'
Azmin: Malaysia not realising full growth potential
López Obrador and Trump speak in aftermath of El Chapo son shootout
Brexit: Johnson sends unsigned letter to EU seeking delay
UK parliament snubs Boris Johnson's Brexit deal
Jho Low's former yacht back on the market again
Mexico admits freeing Chapo's son after 'badly planned' operation
Mark Carney: Brexit deal will boost flagging global economy
Rahman Dahlan wants Guan Eng to step down as finance minister
Zakir Naik sues Ramasamy, seeks apology
Student who attacked police faces life in prison
Police Raid Raises Threat Bomb
in ASEAN, Asia News, Featured, Malaysia, News, Politics
Home Asia News ASEAN
Asked to sum up Malaysia in one word, Tun Dr Mahathir Mohamad gave it as "multiracial" on radio show Hot.Fm this morning.
The 94-year-old prime minister who continues to be the centre of global attention after returning to power with a new political coalition last year was on air for about 1.5 hours, speaking on a diverse range of topics that covered his childhood during the Japanese Occupation during World War Two to his well-known love for cars, mobile phone usage and cash in his wallet.
While books have been written about Dr Mahathir's humble beginnings from selling goreng pisang and his rise to the government's highest office, the Kedahan opened up with less commonly-known tidbits today.
Malaysia to push for Asian Tiger status
Dr M: we will study all offers for PLUS Malaysia takeover
Azmin to present Malaysia's economic direction at UN General Assembly
Dr M moots Japanese university branch campuses in Malaysia
PM: Malaysia proud of Tabung Haji's achievements
Bully victim
Dr Mahathir recalled being bullied by a schoolmate who was also a neighbour's son when a child and helping his family make ends meet during war-time Malaya.
"My parents had no pension and at the time the Japanese were around, many people were out of jobs including government servants, so I had to work.
"I used to sell balloons too make ends meet. Two cents you get three one cent you get one. This guy Hussein who was a son of my neighbour would bully me and then I'd have to go to the canteen and buy him a meal because I was afraid of him."
Cash he carries on his person
Fast forward to today and money is no longer an issue for Dr Mahathir. When asked by the radio presenters how much cash he has in his wallet, the PM smoothly said with a chuckle: "I have enough money".
"My mother used to tell me even if it's one cent, always have money in your pocket as you'll never know when you need it."
Many handphones
"Phone-wise, I have many as people come and give me new phones, but I only use one. The last time, I took pictures with it was when I visited America.
"Also, I was close with the Nelson Mandela but he has passed on. Nowadays, there aren't many world leaders that are good examples so I'm not particularly close with anyone else."
Like everyone else, Dr Mahathir loves a good song. His favourite are those sung by local talent Tan Sri Teuku Zakaria Teuku Nyak Puteh, better known as P. Ramlee.
The last movie he watched was M for Malaysia, which was directed by his granddaughter Ineza Roussille.
But what he really considers entertainment is driving.
"I've always loved to drive, even from back when I was studying. It's the only thing I can think of that I do for entertainment," said Dr Mahathir.
"I used to drive from Singapore to KL then KL to Alor Setar back in the day in an old rickety car which had a roof made of cloth. It was the only one I could afford as a student. Nowadays, they don't make those anymore.
"Even now on weekends, I still like to drive but at night sometimes my feet hurt. I've driven many types of cars, including Proton which always sends me their new models to test out."
Dr Mahathir said the fastest he's driven was 180 km/h.
When he used to travel from Kedah to Penang in the past, he would make a stopover at roadside stalls to indulge in certain food, like nasi kandar, and beef dishes.
He recalled going to a shop called 'Mira' in Kedah and eating nasi kandar from a roadside stall, favouring the beefy dishes above others.
"When the Maju highway wasn't open yet they let us test the road and I went a but fast maybe more than 180km/h. When my wife is in the car however I won't drive so fast.
"During my travels to and from Kedah and Penang there was this shop on the roadside pavement selling nasi kandar and in the past it wasn't easy to get beef. Maybe in a week we'd eat it once or twice and chicken much the same.
"In fact during the Japanese Occupation, we were only given the small prawns and not the big ones, so I'd usually just add the gravy with the rice and eat it like that."
Friends he's still in touch with
Pushing on 95, Dr Mahathir admitted most of his friends have long since passed on but he still has two friends of whom he tries to keep in touch with. Their names are Khalid Julong and Osman Kuda, both in their nineties as well.
Dr Mahathir also said he always had aspirations to be in politics.
Asked what he felt was important for the next prime minister of Malaysia to have, he said: "Honesty is important and focus on work and not yourself".
"As leaders we have to work harder than those we're leading. They must also be able to settle soft problems."
Malaysia Mail
Tags: Ineza RoussilleMaju highwaynasi kandarTan Sri Teuku Zakaria Teuku Nyak PutehTun Dr Mahathir Mohamad
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text:
\section{Introduction}
\noindent The Erd\H{o}s-Szekeres problem is defined as follows:
\textit{For any integer k, k \ensuremath{\ge}
3, determine the smallest positive integer N(k) such that any planar point set in general position that has at least N(k) points contains k points that are the vertices of a
convex k-gon.}

In 1935 Erd\H{o}s and Szekeres proved the finiteness of $N(k)$ using Ramsey theory \cite{Erdos35acombinatorial}.
There has been a series of improvements to bound the value of $N(k)$ and the current best known bounds
are 2$^{k-2}$+ 1 \ensuremath{\le} $N(k)$ \ensuremath{\le} $\left(_{k-2}^{2k-5}\right)$+1 \cite{Erdos78somemore,Kleitman98findingconvex,Toth04theerdos-szekeres}.
 Erd\H{o}s and Szekeres conjectured that the current lower bound is tight. This conjecture has been proved for $k$ \ensuremath{\le} $6$.
$N(4)=5$ was shown by Klein and $N(5)= 9$ was shown by Kalbfeisch $et.al.$ \cite{kalbfeisch}.
 $N(6)=17$ has been proved by Szekeres and Peters using an combinatorial model of planar configurations  \cite{Szekeres_computersolution}.
See survey \cite{Morris00theerdos-szekeres,newsurvey} for a detailed description about the history of the problem  and its variants.

Erd\H{o}s asked the empty convex $k$-gon problem which is defined as follows:
\textit{ For any integer k, k \ensuremath{\ge}
3, determine the smallest positive integer $H(k)$ such that any planar point set in general position that has at least $H(k)$ points contains k points
 that are the vertices of a
an empty convex $k$-gon, i.e., the vertices of a convex $k$-gon containing
no points in its interior.}

It is easy to see that $H(4) = 5$. $H(5) = 10$ was proved by Harboth \cite{Harborth79konvexefunfecke}. Gerken \cite{Gerken05onempty} and Nicholes \cite{1340381}  proved independently the existence of a empty hexagon.
  The current best bounds on $H(6)$ are $30 \leq H(6) \leq 463$ \cite{kos,overmars}. Horton showed that $H(k)$ does not exist for any $k$, $k$ \ensuremath{\ge}
$7$ \cite{Horton83setswith}.

The Erd\H{o}s-Szekeres problem has been studied in higher dimensions \cite{Erdos35acombinatorial,Erdos78somemore,JOUR}.
Other related variants that are well studied are the convex $k$-gon with specified number of
interior points \cite{Avisonthe1,on,Fevens04improvedlower,Hosono2005201,1667532}
 and the chromatic variant  \cite{Devillers03chromaticvariants,nonconvex}.

 We introduce the following two player game variant of Erd\H{o}s-Szekeres problem:

  \textit{Consider a two player game where each player playing in alternate turns, place points in the plane.
 The objective of the game is to avoid the formation of the convex k-gon among the placed points.
The game will end when a convex k-gon is formed and the player who placed the last point loses the game.}

In the game we assume that each player has infinite computational resources and hence place their points in optimal manner.

 Since $N(k)$ is finite, we know that the game will end at $N(k)$ number of steps. Can the game end before $N(k)$ steps?
 Define $N_{G}(k)$ as the minimum number of steps before the game ends. In this paper we focus on finding the exact value of $N_{G}(k)$.
We denote the player who plays first as player 1 and the player who plays second as player 2.

We also consider the two player game for the empty convex $k$-gon and correspondingly define $H_{G}(k)$.
 It is easy to see that  $N_{G}(3)=H_{G}(3)=3$, $N_{G}(4)= H_{G}(4)=5$.

\subsection*{Results}
In this paper, we focus on the Erd\H{o}s-Szekeres two player game for $k = 5$. We show a winning strategy  for player 2 and prove that $N_{G}(5)=9$ and $H_{G}(5)=9$.
i.e., the game will end in the 9th step.

We consider convex layer configurations at each step and give a strategy for player 2 such that the game will
reach a specific set of configurations until the 8th step and finally we argue in the 9th step that a convex 5-gon or an empty convex 5-gon is formed.

\subsection*{Organization of the paper}
Section 2 contains the preliminaries and definitions that we will be using in the rest of the paper. Section 3  describes the proofs that
 the empty convex 5-gon game and convex 5-gon game ends in 9th step and player 2 wins
the game.

\section{Preliminaries and Definitions}

We assume in the rest of this paper that our point set $P$ is in general position, i.e., no 3 points of the point set are collinear.
We denote the convex hull of a point set $P$ as $conv(P)$ and the vertices of $conv(P)$ as $CH(P)$.

\begin{defn}\label{convex k-gon}\textbf{Convex k-gon:}
  is a convex polygon with k vertices.
\end{defn}

\begin{defn}\label{empty convex k-gon}\textbf{Empty Convex k-gon:}
  is a convex k-gon with
no points in its interior.
\end{defn}

\begin{defn}\label{Points in convex position}\textbf{Points in convex position:}
A point set $P$ is said to be in convex position if $CH(P)= P$.
\end{defn}

\begin{defn}\label{convex layers}\textbf{Type of a point set P:}
A point set $P$ in of $type (i_{1},i_{2},...i_{k})$, $|P|=\sum|i_{k}|$, if $P_{1}= CH(P)$ is of size $i_{1}$, $P_{2}= CH(P \setminus P_{1})$ is of
size $i_{2}$ \dots
\end{defn}

The $type$ of point set $P$ describes the sizes of the different
convex layers of $P$. We denote $P_{1}$ as the first convex layer and $P_{2}$ as the second convex layer.

\begin{defn}\label{using}\textbf{U(i,j) of point set P:}
 Point set having $i$ points of the first convex layer of $P$
and $j$ points of the second convex layer of  $P$.
\end{defn}

\begin{defn}\label{type1}\textbf{Type 1 Beam:}
  $A:BC$ denotes the region of the plane  formed by deleting triangle $ABC$ from the convex region in the plane
bounded by the rays $\overrightarrow{AB}$ and $\overrightarrow{AC}$ (see figure \ref{typa}).
\end{defn}

\begin{defn}\label{type2}\textbf{Type 2 Beam:}  $AB:CD$ denotes the region of the plane formed by deleting convex 4-gon $ABCD$
from the convex region in the plane
 bounded by the segment $\overline{AB}$ and the rays $\overrightarrow{AD}$ and $\overrightarrow{BC}$ (see figure \ref{typb}).
\end{defn}

\begin{figure}[ht]
\begin{minipage}[b]{0.5\linewidth}
  \centering
   {\includegraphics[width=0.5\textwidth]{type1beamv1.eps}}
   \hspace{0.3\textwidth}
   \caption{Type 1 beam}
    \label{typa}
\end{minipage}
\begin{minipage}[b]{0.5\linewidth}
  \centering
   {\includegraphics[width=0.5\textwidth]{type2beamv1.eps}}
   \hspace{0.3\textwidth}
   \caption{Type 2 beam}
    \label{typb}
\end{minipage}
\end{figure}
 \begin{figure}
  \centering
  {\label{4gonregionse}\includegraphics[width=0.5\textwidth]{conve4.eps}}
  \hspace{0.1\textwidth}
  \caption{Divison of convex 4-gon into regions}
\end{figure}

 Let $A,B,C,D$ be 4 points in convex position.
\begin{defn}\label{coongrm}\textbf{Regions of empty convex 4-gon:}
The $ABCD$ convex 4-gon  divides the plane into 4 types of regions $I,O,S,Z$ (see $figure$ 3).
$O$ region is the region such that any point in it along with $ABCD$  forms a convex 5-gon.
$I$ and $Z$ regions are the regions such that any point in these regions along with $ABCD$ forms a point set of $type(4,1)$. $I$ region is outside the convex 4-gon and
 $Z$ region is inside the convex 4-gon.
$S$ region is the region such that any point in it along with $ABCD$  forms a point set of $type(3,2)$.
\end{defn}

We define the following set of point configurations that will be used in our proofs (see $figure$ \ref{tree}).

\begin{defn}\label{config4}\textbf{Configuration 4:}
A 4 point set forming a parallelogram.
\end{defn}
\begin{defn}\label{config5.1}\textbf{Configuration 5.1:}
A 5 point set of $type(4,1)$ where the 4 points in the first convex layer form a parallelogram.
\end{defn}
\begin{defn}\label{config5.2}\textbf{Configuration 5.2:}
A 5 point set of $type(4,1)$ where the 3 points in the first convex layer with the 1 point in the interior form a parallelogram.
\end{defn}

 \begin{figure}
  \centering
  {\includegraphics[width=1.0\textwidth]{urgent1.eps}}
  \hspace{1.0\textwidth}
  \caption{Game tree for the convex 5-gon and empty convex 5-gon}
\label{tree}
 \end{figure}

\begin{defn}\label{config6.1}\textbf{Configuration 6.1:} A  6 point set  of $type(4,2)$ where
4 points in the first convex layer form a parallelogram and the 2 points inside the parrallelogram are symmetrically placed in opposite triangles formed by diagonals of parallelogram.
\end{defn}

\begin{defn}\label{config6.2}\textbf{Configuration 6.2:} A  6 point set of $type(4,2)$ where
4 points in the first convex layer form a trapezoid and the 2 points inside the trapezoid are symmetrically placed
 in  opposite triangles formed by the diagonals.
\end{defn}

\begin{defn}\label{config7.1}\textbf{Configuration 7.1:}
 A 7 point set of $type(3,4)$  where  the 3 points of the first convex layer are in  different $I$ regions
of the  parralleogram formed by the 4 points in the second convex layer.
\end{defn}

\begin{defn}\label{config7.2}\textbf{Configuration 7.2:}
 A 7 point set of $type(4,3)$  such that it does not have an empty convex 5-gon.
\end{defn}
 \begin{defn}\label{config8}\textbf{Configuration 8:}
 An  8 point set of $type(4,4)$  where  the 4 points of the first convex layer are placed such that each point lies in  different $I$ region
of the  convex 4-gon of the second convex layer.
\end{defn}

Note that the above configurations of $i$ points  $4 \leq i \leq 8 $ do not contain a empty convex 5-gon and all configurations, except configuration 7.2
and 8 do not contain convex 5-gon.
\section{Game for the empty convex 5-gon and convex 5-gon}
 In this section, we show that the two player game for the empty convex 5-gon  and the convex 5-gon  ends in  9 moves and the second player has a winning strategy.
\subsection*{Overview of our proof and player 2's strategy to win the empty convex 5-gon game and the convex 5-gon game.}

In our game, Player 1 plays in the odd steps (1st, 3rd, \dots) and Player 2 plays in the even steps (2nd, 4th, \dots).
 In player 1's turn, we argue that any point added without forming an convex 5-gon or empty convex 5-gon will always result
in specific configurations. In player 2's turn we show a feasible region where if the point is placed will result in specific configurations that are
favorable for player 2.

We now describe the winning strategy for player 2:
Player 2  will  place the point in the 4th step such that the resultant point set forms a parallelogram (configuration 4).
In the  6th step, we show that there exists a feasible region in both configuration 5.1 and 5.2 where the 6th point is placed, so that it will reach
configuration 6.1 or configuration 6.2. Similarly in the 8th step we show that there will exist a feasible region in configuration 7.1 and 7.2  where the 8th point
is placed such that the resultant point set is configuration 8.

In player 1's turn we show that any point added by player 1 without forming an convex k-gon or empty convex k-gon results in configuration 5.1 or 5.2 (5th step) and in configuration 7.1 or 7.2 (7th step).

Finally, we argue that any point added to the configuration 8 will result in the formation of a convex 5-gon or empty convex 5-gon and player 2 will always win
 in the 9th step.

Thus, any convex 5-gon/empty convex 5-gon game follows a path in the game tree shown in $figure$ \ref{tree}.
The proofs for the convex 5-gon game and the empty convex 5-gon are similar, so we have combined them.

In the odd step (player 1's turn to place the point) the feasible regions that are formed in the convex 5-gon game
 are a subset of the feasible regions that are formed in the empty convex 5-gon game. This is because the regions which
are not covered by the O regions of the convex 4-gons are a subset of the regions which are not covered by the O regions of the empty convex 4-gons.
 So in the odd step we give the proof for the empty convex 5-gon game and it is sufficient for both the games.

In the even step (player 2's turn to place the point) we give the proof for the convex 5-gon game by showing a feasible region where player 2
places the point and this is sufficient for both the games because a feasible region in the convex 5-gon game is also a feasible region in the empty convex
5-gon game.

In the empty convex 5-gon game, we give a separate proof for point configurations which have a convex 5-gon that is not empty.

\subsection{Proof for  $H_{G}(5)=9$ and $N_{G}(5)=9$}
We make the following observations on the number of points that can contained in type 1 and type 2 beams without forming an empty convex 5-gon. We will assume that the points in the beam are in convex position with
the points that form the beam. Thus, if the point set does not contain a convex 5-gon, type 1 beam has atmost 1 point and type 2 beam does not contain any point.

\begin{figure}
  \centering
  {\includegraphics[width=0.3\textwidth]{parallel.eps}}
  \hspace{0.1\textwidth}
  \caption{Divison of parallelogram regions}
  \label{parrallelogram}
\end{figure}

\begin{lem}
\label{lem1}
The game for the convex 5-gon and the empty convex 5-gon will always reach either configuration 6.1 or configuration 6.2 at the end of 6th step.

\end{lem}
\begin{proof} A triangle is formed by the first 3 points of the game. The 4th point is placed in such a manner that the resultant point set forms a parallelogram $ABCD$ (configuration 4).
For the 5th step, we divide the regions of the parallelogram into $I, O, Z$ regions as shown in $figure$  \ref{parrallelogram}.
If a point is placed in the $O$ region it forms an empty convex 5-gon with the four points of the parallelogram.
 So the only feasible regions where a point is placed  are the $I$ regions and $Z$ region.

\begin{figure}[ht]
\begin{minipage}[b]{0.5\linewidth}
  \centering
   {\includegraphics[width=0.65\textwidth]{con51v1.eps}}
   \hspace{0.18\textwidth}
   \caption{Configuration 5.1}
\label{config5.1}
\end{minipage}
\begin{minipage}[b]{0.5\linewidth}
  \centering
   {\includegraphics[width=0.67\textwidth]{con52v1.eps}}
   \hspace{0.3\textwidth}
   \caption{Configuration 5.2}
\label{config5.2}
\end{minipage}
\end{figure}

If the 5th point, say $E$, is placed in the interior of the parallelogram i.e., $Z$ region, the resultant point set forms configuration 5.1 (see $figure$ \ref{config5.1}).
If $E$ is placed in $I$ region the resultant point set forms configuration 5.2 (see $figure$ \ref{config5.2}).

\begin{figure}[ht]
\begin{minipage}[b]{0.5\linewidth}
  \centering
   {\includegraphics[width=0.67\textwidth]{parallel1v1.eps}}
   \hspace{0.3\textwidth}
   \caption{Configuration 6.1}
\label{config6.1}
\end{minipage}
\begin{minipage}[b]{0.5\linewidth}
  \centering
   {\includegraphics[width=0.68\textwidth]{parallel2v1.eps}}
   \hspace{0.3\textwidth}
   \caption{Configuration 6.2}
\label{config6.2}
\end{minipage}
\end{figure}

In the 6th step we show a feasible region in configuration 5.1 and configuration 5.2  where a point is added
such that the resultant point set formed is either configuration 6.1 or configuration 6.2.

If the 5 point set formed is  configuration 5.1, then the 6th point, say $F$ is placed
in a triangle (formed by the diagonals of the parallelogram)
 that is opposite to the triangle that has $E$ such that $EF$ is parallel to $AD$ and this forms configuration 6.1
(see $figure$ \ref{config6.1}).

Now we argue for configuration 5.2. Let us suppose that $E$ has been placed in $I_{1}$ region (see $figure$ \ref{config5.2}).
The 6th point, say $F$, is placed in $I_{8}$ region  such
that $EF$ is parallel to $AD$ (see $figure$  \ref{config6.2}). Let us call this as $\left\{ I_{1},I_{8}\right\}$
point placement. By symmetry $ \left\{ I_{2},I_{3}\right\} ,\left\{ I_{4},I_{5}\right\} ,\left\{ I_{6},I_{7}\right\}$
 point placements are similar and they form configuration 6.2.
\end{proof}

\begin{lem}
\label{lem2}
  Any point added to either configuration 6.1 or configuration 6.2 without forming an convex 5-gon or an empty convex 5-gon,
results in either configuration 7.1 or configuration 7.2.
\end{lem}
\begin{proof}
We consider 2 cases corresponding to configuration 6.1 and configuration 6.2. We denote the regions of configuration 6.1 and 6.2  as $O$ region
if it is infeasible (point added in this region forms an empty convex 5-gon) and $I,S,Z$ region if there exists a feasible region in their interior.

\begin{figure}
  \centering
   {\includegraphics[width=0.42\textwidth]{con6.1v2.eps}}
   \hspace{0.1\textwidth}
   \caption{Configuration 6.1 divided into regions}
\label{p4}
\end{figure}

\textbf{Case 1 (configuration 6.1):} Let us consider all the empty convex 4-gons of configuration 6.1 (see $figure$ \ref{p4}).
The regions that are not covered by the $O$ regions of the empty convex 4-gons are precisely the regions where a point can be
 added without forming a empty convex 5-gon.

The convex 4-gons that are formed by $U(4,0),U(3,1)$ of configuration 6.1 are not empty, so they are not considered.
 The empty convex 4-gons are of the form  $U(2,2)$ of configuration 6.1.

\textit{U(2,2) of configuration 6.1:}
$EFDA, EFCB, BEDF, AECF$ are the empty convex 4-gons. Consider $EFDA$ empty convex 4-gon. The regions that are covered by the $O$ regions of $EFDA$ are $O_{1},O_{7},O_{8},O_{9},O_{10}$.
 Similarly the regions that are covered by the $O$ regions of $EFCB$ empty convex 4-gon are $O_{2},O_{3},O_{4},O_{5},O_{6}$.
  The $O$ regions of $BEDF, AECF$ empty convex 4-gons does not cover the entire region of $Z$ and $I_{k}$ where $k=1,2,3,4$.
Thus, the only feasible regions of configuration 6.1 are portions of $I$ and $Z$ region.
 From $figure$ \ref{p4}, we can verify that
region $I^{*}$ of $I_{1}$ and $Z^{*}$ of $Z$ is not covered by any of the $O$ regions of
the empty convex 4-gons. Figure \ref{p4} shows $I^{*}$ only in one  $I$ region and $Z^{*}$ in $Z$ region.
Symmetrically, there are feasible regions $I^{*}$ in the other $I$ regions and $Z^{*}$ in $Z$ region. Any point added in $I^{*}$ or $Z^{*}$ region will result in configuration 7.2.
\newline
\textbf{Case 2 (configuration 6.2):} Let us consider the empty convex 4-gons of configuration 6.2 (see $figure$ \ref{1o}). The convex 4-gons that are formed by $U(4,0),U(3,1)$ of configuration 6.2 are not empty, so they are not considered.
 The empty convex 4-gons are of the form  $U(2,2)$ of configuration 6.2 (see $figure$ \ref{1o}).
 \begin{figure}
  \centering
   {\includegraphics[width=0.6\textwidth]{config6.2v3.eps}}
   \hspace{0.1\textwidth}
   \caption{Configuration 6.2 divided into regions}
\label{1o}
\end{figure}

\textit{U(2,2) of configuration 6.2:}
$EFCD, ABFE, AFCE, BEDF$ are the empty convex 4-gons.
Consider $EFCD$ empty convex 4-gon. The regions that are completely covered by the $O$ regions of $EFCD$ are
$O_{2},O_{6},O_{1}$.
 Similarly the regions covered by the $O$ regions of $ABFE$ empty convex 4-gon are $O_{3},O_{5},O_{4}$.
The O-regions of  $AFCE$,$BEDF$ empty convex 4-gons does not cover the entire regions of  $Z$ and $I_{k}$ where $k=1,2,3,4$.
Thus, the only feasible regions of configuration 6.2 is $S_{1},S_{2}$ and portions of $I$ and $Z$ regions.
From $figure$ \ref{1o}, we can verify that region $I^{*}$ of $I_{2}$  is not covered by any of the $O$ regions of these convex 4-gons.
Figure \ref{1o} shows $I^{*}$ only in one  $I$ region and $Z^{*}$ in $Z$ region.
Symmetrically, there are feasible regions $I^{*}$ in the other $I$ regions and $Z^{*}$ in $Z$ region.

If the point is placed in $I^{*}$ of $I_{k}$ where $k=1,2,3,4$ or $Z^{*}$ of $Z$,
 the resultant point set is configuration 7.2.
If the point is placed in $S_{1}$ or $S_{2}$ the resultant point set is configuration 7.1.
\end{proof}

Now we prove that given a set of 7 points in configuration 7.1 or 7.2 there exists a feasible region
  where a point is added such that the resultant point set is configuration 8.

\begin{lem}
\label{le3}
There exists a feasible region in configuration 7.1 and 7.2 such that a point added in the feasible regions results in configuration 8.
\end{lem}
\begin{proof}
We consider 2 cases corresponding to configuration 7.1 and configuration 7.2.
 \begin{figure}
  \centering
   {\includegraphics[width=0.45\textwidth]{sp34v3.eps}}
   \hspace{0.1\textwidth}
   \caption{Configuration 7.1}
\label{lq1}
\end{figure}

\textbf{Case 1 (configuration 7.1):}
 Let us consider all the convex 4-gons of configuration 7.1. The regions that are
 not covered by the $O$ regions of convex 4-gons are precisely the regions where a point is added without forming a convex 5-gon. Consider the region $I^{*}$ as shown in $figure$ \ref{lq1}. We will show that this is a feasible region.
 The convex 4-gons are of the form $U(2,2),U(1,3),U(0,4)$ of configuration 7.1.

\textit{U(0,4) of configuration 7.1:}
$ABCD$ is the only convex 4-gon of this type and has $I^{*}$ in its $I$ region.

\textit{U(1,3) of configuration 7.1:}
 $EBDA,FDBC,ADGC,ABGC,DCEB,ABDF$ are the convex 4-gons of this type and these have $I^{*}$ in their $I$ regions.

\textit{U(2,2) of configuration 7.1:}
$FCDG,EACF,ABGF,BDFE,ADGF,FCBG$ are the convex 4-gons of this type and these have $I^{*}$ in their $I$ regions.
 Hence $I^{*}$ is feasible region where a point is added and the resultant point set that is formed is configuration 8.

\textbf{Case 2 (configuration 7.2)}: Configuration 7.2 is a point set of (4,3) convex layer configuration without a convex 5-gon.
First, we give the following characterization of configuration 7.2:
 \begin{figure}
  \centering
   {\includegraphics[width=0.3\textwidth]{drawing.eps}}
   \hspace{0.1\textwidth}
   \caption{Divison of triangle into regions}
\label{12}
\end{figure}
\textit{An $I$ region or an $O$ region of the inner triangle cannot have more than 1 point (see $figure$ \ref{12})}.

 Consider an $O$ region. If an $O$ region has 2 points then there is an empty convex 5-gon being formed by these 2 points with the 3 points of the triangle.
 Hence an $O$ region cannot have more than 1 point.
Consider an $I$ region. If an $I$ region has 2 points then no other $I$ region or
the adjacent $O$ regions has any points because the 2 points in this $I$ region along with the third point in the $I$ region
 or in the adjacent $O$ region will form an empty convex 5-gon with the side of the triangle. Thus if an $I$ region has  2 points then the $O$ region that is
opposite has to have the other 2 points which leads to the formation of an empty convex 5-gon.

Based on the above constraints, the only
possible (4,3) convex layer configuration is $(I,I,O,O)$ (2 points each in different $I$ regions, 2 points each in different $O$ regions).

Let us assume that point $G$ lies in region $X_{1}$. The case when the point $G$ is in $X_{2}$ can be argued in a similar fashion (see $figure$ \ref{lq5},\ref{lq7}).
 We have 2 cases corresponding to $EBG$ being a anti-clockwise turn or clockwise turn.

Type 1 (when $G$ lies in $X_{1}$ and $EBG$ is a anti-clockwise turn):
 We will show that $I^{*}$ (triangle region bounded  by the  $CG$,$FA$ and $EA$) is a feasible region (see $figure$ \ref{lq4}).
Consider the  convex 4-gons of this configuration.
The convex 4-gons are either of the type $U(2,2)$ ,$U(1,3)$ or $U(3,1)$.
\begin{figure}
  \centering
   {\includegraphics[width=0.53\textwidth]{config722v2.eps}}
   \hspace{0.1\textwidth}
    \caption{Configuration 7.2 (I,I,O,O) G is in X1 and EBG is anticlockwise turn}
\label{lq4}
\end{figure}

\textit{U(1,3) of Type1 Configuration 7.2 (I,I,O,O):}
$EACB,FABC$ are the 2 convex 4-gons of this type. $I^{*}$ region is in the interior of $EACB$ and it is in the $I$ region of $FABC$.
Thus $I^{*}$ is feasible for these convex 4-gons.

\textit{U(2,2) of Type1 Configuration 7.2 (I,I,O,O):}
$EABH,FACG,ECGB,EACH$ $,FABG,BCFE,AHCF,EBGA$ and $HBFC$ or $AFBH$ based upon the position of $H$ in the $I$ region are the convex 4-gons of this type. It
is easy to see that
 $I^{*}$ region is in the interior of $EABH,EACH,BCFE,EBGA$ and $I^{*}$ region is in the $I$ region of $FACG,ECGB,FABG,AHCF$. If $FAH$ is an anti clockwise turn and
$FCH$ is clockwise turn,  then either $HBFC$ or $AFBH$ is the  convex 4-gon formed based upon the position of $H$ in the
$I$ region of the triangle and both these  convex 4-gons have $I^{*}$ in their $I$ region. Thus $I^{*}$ is feasible for these convex 4-gons.

\textit{U(3,1) of Type1 Configuration 7.2 (I,I,O,O):}
$EAGH,FAHG ,FBHG,ECGH,$ $EHCF,EBGF$ are the convex 4-gons of this type and these 4-gons have $I^{*}$ either in their $I$ region or in their $Z$ region.
If $FAH$ is an clockwise turn  then $EHAF$ is a convex 4-gon of this type having  $I^{*}$ region inside.
If $FAH$ is an anti clockwise turn and $FCH$ is anticlockwise turn then $FCHG$ is the  convex 4-gon is of this type and has $I^{*}$ in its $I$ region.
Thus $I^{*}$ is feasible for these convex 4-gons.
Any point added to $I^{*}$ results in configuration 8.

Type 2 (when $G$ lies in $X_{1}$ and $EBG$ is a clockwise turn):
Consider the  convex 4-gons of this configuration.
The convex 4-gons are either of the type $U(2,2)$ ,$U(1,3)$ or $U(3,1)$ (see $figure$ \ref{lq6}).

\textit{U(3,1) of Type2 Configuration 7.2 (I,I,O,O):}  The convex 4-gons are the same as U(3,1) of Type1 Configuration 7.2 (I,I,O,O)
except that we have $EBGH$ instead of $EBGF$. $EBGH$ has $I^{*}$  in its $I$ region.

\textit{U(1,3) of Type2 Configuration 7.2 (I,I,O,O):}
This is same as U(1,3) of Type1 Configuration 7.2 (I,I,O,O).

\textit{U(2,2) of Type2 Configuration 7.2 (I,I,O,O):}
This is a subset of U(2,2) of Type1 Configuration 7.2 (I,I,O,O).

For the empty convex 5-gon game, the following is a valid (4,3) configuration  containing a non empty convex 5-gon.
\begin{figure}
  \centering
   {\includegraphics[width=0.53\textwidth]{config721v2.eps}}
   \hspace{0.1\textwidth}
   \caption{Configuration 7.2 (I,I,O,O) G is in X1 and EBG is clockwise turn}
\label{lq6}
\end{figure}

\begin{figure}
  \centering
   {\includegraphics[width=0.53\textwidth]{config723v2.eps}}
   \hspace{0.1\textwidth}
   \caption{Configuration 7.2 (I,I,O,O) G is in X2 and EAG is an clockwise turn}
\label{lq5}
\end{figure}
\begin{figure}
  \centering
   {\includegraphics[width=0.53\textwidth]{config724v2.eps}}
   \hspace{0.1\textwidth}
   \caption{Configuration7.2 (I,I,O,O) G is in X2 and EAG is anticlockwise turn}
\label{lq7}
\end{figure}
(I,O,O,O) (3 points each in different $O$ regions, 1 point in a $I$ region): Let us consider all the empty convex 4-gons of
this configuration (see $figure$ \ref{label13}).
 The regions that are not covered by the $O$ regions of empty convex 4-gons are precisely the regions where a point is added without forming an empty convex 5-gon.
We will show that there exists a region $I^{*}$ in these feasible regions where a point is placed forming configuration 8.
The convex 4-gons that are formed by $U(4,0)$ of configuration 7.2 are not empty, so they are not considered.
 The remaining empty convex 4-gons are of the form  $U(2,2),U(1,3),U(3,1)$ of configuration 7.2.
It is important to note that $GHE$ should be an anticlockwise turn and $GFE$ a clockwise turn, otherwise we do not have convex 4-gon in the first convex layer.

\begin{figure}
  \centering
   {\includegraphics[width=0.47\textwidth]{a43v2.eps}}
   \hspace{0.2\textwidth}
   \caption{Configuration 7.2 (I,O,O,O)}
    \label{label13}
\end{figure}
\textit{U(3,1) of configuration 7.2 (I,O,O,O):}
$EHAF$ is the empty convex 4-gon of this type and it has $I^{*}$ in its $I$ region. Thus $I^{*}$ is feasible for these convex 4-gons.

\textit{U(1,3) of configuration 7.2 (I,O,O,O):}
$FACB,HABC,ACGB$ are the empty convex 4-gons
 of this type. $FACB,HABC$ have $I^{*}$ in there $I$ regions and $ACGB$
has $I^{*}$ inside. Thus $I^{*}$ is feasible for these convex 4-gons.

\textit{U(2,2) of configuration 7.2 (I,O,O,O):}
$EABF,EHCA$ are the empty convex 4-gons of this type and these have $I^{*}$ in there $I$ regions. Thus $I^{*}$ is feasible for these convex 4-gons.
Any point added to $I^{*}$ results in configuration 8.

\end{proof}
We call a point set as bad configuration for the convex 5-gon game if the point set has no convex 5-gon and any point added to the point set forms an convex 5-gon.
Similarly, we can define bad configuration for the empty convex 5-gon game. Note that the game ends in the $i$th step only if the point set reached in
the $i-1$th step is a bad configuration.

We now argue that the game will always reach the 9th step, i.e., there is no possibility for the game to end earlier.
 To prove this we show that there does not exist point set
 with $2k$ points where $k=2,3$ that are bad configurations.
 We do not consider point set with $2k+1$ points, $k=2,3$, because it is player 2's turn to form such a point set
and  even though there are such point sets which are bad configurations, player 2 is able to avoid them in the game by ensuring that configuration 5 is reached
in the 5th step and configuration 7.1 or configuration 7.2 is reached in the 7th step. By lemma \ref{lem1}, \ref{le3} there exists feasible region to
place points and hence these configurations are not bad configurations.

It is easy to see that no point set with 4 points are bad configurations.

\begin{lem}
\label{added}
 There are no point set  with 6 points that are bad configurations.
\end{lem}
\begin{proof}

When the point set contains 6 points it is easy to see that (3,3) and (4,2) are the only
 valid convex layer configurations for the  convex 5-gon game and (3,3), (4,2), (5,1)
are the valid convex layer configurations for the empty convex 5-gon game.
To show that a (3,3), (4,2) convex layer configurations are not bad configurations we show a feasible region where a point is added without
forming an convex 5-gon.
To show that a (5,1) convex layer configuration is not a bad configuration we show a feasible region where a point is added without
forming an empty convex 5-gon.
\begin{figure}[ht]
\begin{minipage}[b]{0.53\linewidth}
  \centering
   {\includegraphics[width=0.5\textwidth]{lastproof21v2.eps}}
   \hspace{0.1\textwidth}
   \caption{Case 1 of (4,2) configuration}
\label{lastproof21o}
\end{minipage}
\begin{minipage}[b]{0.53\linewidth}
  \centering
   {\includegraphics[width=0.5\textwidth]{lastproof22v3.eps}}
   \hspace{0.1\textwidth}
   \caption{Case 2 of (4,2) configuration}
\label{lastproof21s}
\end{minipage}
\end{figure}
\begin{figure}
  \centering
   {\includegraphics[width=0.25\textwidth]{lastproof23v3.eps}}
   \hspace{0.1\textwidth}
   \caption{Case 3 of (4,2) configuration}
\label{lastproof21a}
\end{figure}

\textit{(4,2) Convex layer configuration:} Consider the line joining $EF$  of the second convex layer (see $figure$ \ref{lastproof21o}). This line divides the first convex layer into 2 parts. If either of the parts contains
 3 points of the first convex layer then they form an empty convex 5-gon with the 2 points of the second convex layer.
 So the only other possible option is that both the parts contain 2 points.

Based upon the position of $E,F$ in the diagonal triangles of the first convex layer, (4,2) convex layer configurations are divided into 3 cases.

Case 1:  $E,F$ are in opposite triangles, see $figure$ \ref{lastproof21o}.

Case 2:  $E,F$ are in same triangle, see $figure$ \ref{lastproof21s}.

Case 3:  $E,F$ are in adjacent triangles, see $figure$ \ref{lastproof21a}.

In each of the above cases a feasible region $K$ exists where a point can be added without forming a convex 5-gon (see shaded
region in $figure$ \ref{lastproof21o}, \ref{lastproof21s}, \ref{lastproof21a}).
Thus (4,2) convex layer configurations are not bad configurations.

\textit{(3,3) Convex layer configuration:}

 We make the following observations on the placement of the three points of the first convex
layer in the $I$ and $O$ regions of the triangle formed by the second convex layer (see $figure$ \ref{12}). If an $I$ region has 3
 points then the triangle in the second convex layer will not be contained in the triangle of the
first convex layer. Hence an $I$ region cannot have 3 points. Similarly an $O$ region also cannot have 3 points. If an $O$ region has
 2 points then the opposite $I$ region has to have the third point. In this case, there is an empty convex 5-gon being formed by
these 2 points with the 3 points of the triangle. Hence an $O$ region cannot have more than 1 point.

\begin{figure}[ht]
\begin{minipage}[b]{0.48\linewidth}
  \centering
   {\includegraphics[width=0.53\textwidth]{lastproof331v2.eps}}
   \hspace{0.1\textwidth}
   \caption{(I,I,I) configuration}
\label{lastproof1}
\end{minipage}
\begin{minipage}[b]{0.5\linewidth}
  \centering
   {\includegraphics[width=0.6\textwidth]{lastproof332v2.eps}}
   \hspace{0.1\textwidth}
   \caption{(O,O,O) configuration}
\label{lastproof2}
\end{minipage}
\end{figure}

\begin{figure}[ht]
\begin{minipage}[b]{0.5\linewidth}
  \centering
   {\includegraphics[width=0.72\textwidth]{lastproof333v2.eps}}
   \hspace{0.1\textwidth}
   \caption{(I,I,O) configuration}
\label{lastproof3}
\end{minipage}
\begin{minipage}[b]{0.5\linewidth}
  \centering
   {\includegraphics[width=0.72\textwidth]{lastproof334v2.eps}}
   \hspace{0.1\textwidth}
   \caption{(O,O,I) configuration}
\label{lastproof4}
\end{minipage}
\end{figure}

The only possible (3,3) point configurations are the following:

$(I.I,I)$ (3 points in different $I$ regions) $figure$ \ref{lastproof1}.

$(O,O,O)$ (3 points in different $O$ regions) $figure$ \ref{lastproof2}.

$(I,I,O)$ (2 points are in different $I$ regions and 1 point in an  $O$ region) $figure$ \ref{lastproof3}.

$(O,O,I)$ (2 points are in different $O$ regions and 1 point in an  $I$ region) $figure$ \ref{lastproof4}.

$(2I,O)$ (2 points in an $I$ region and 1 point in the opposite $O$ region) $figure$ \ref{lastproof5}.

In each of the above cases a feasible region $K$ exists where a point can be added without forming a convex 5-gon (see shaded
region in $figure$ \ref{lastproof1}, \ref{lastproof2}, \ref{lastproof3}, \ref{lastproof4}, \ref{lastproof5}).
Thus (3,3) convex layer configurations are not bad configurations.

\textit{(5,1) Convex layer configuration:}
A feasible region $K$ exists where a point can be added without forming a empty convex 5-gon (see shaded region in $figure$ \ref{lastproof31}).

\end{proof}

\begin{figure}[ht]
\begin{minipage}[b]{0.49\linewidth}
  \centering
  {\includegraphics[width=0.75\textwidth]{lastproof335v2.eps}}
   \hspace{0.1\textwidth}
   \caption{(2I,O) configuration}
\label{lastproof5}
\end{minipage}
\begin{minipage}[b]{0.51\linewidth}
  \centering
  {\includegraphics[width=1.0\textwidth]{lastproof336v2.eps}}
   \hspace{0.1\textwidth}
   \caption{(5,1) configuration}
\label{lastproof31}
\end{minipage}
\end{figure}

\begin{thm}
 The convex 5-gon game always ends in the 9th step.
\end{thm}
\begin{proof} The game does not end in the 5th step because no 4 point sets  are bad configurations. By lemma \ref{lem1}, the game will reach
either configuration 6.1 or 6.2. Since all 6 point sets are not bad configurations(lemma \ref{added}), the game will not end in the 7th step.
 By lemma \ref{lem2} the game will reach either configuration 7.1 or 7.2. By lemma \ref{le3}, the game will reach configuration 8 and finally
since $N(5)=9$ the game ends in the 9th step and
player 2 wins the game.
\end{proof}

We will now show that any point added to configuration 8 forms an empty convex 5-gon and hence the empty convex 5-gon game also ends
in the 9th step. First we prove the following lemma.

\begin{lem}
\label{lem4}
A (4,3,2) convex layer configuration has an empty convex 5-gon.
 \end{lem}
\begin{proof}
Let $F,G,H,J$ be the 4 points of $CH(P)$. Let $A,B,C$ be the points
 in the second convex layer and $D,E$ be the points of the inner most convex layer  (see $figure$ \ref{222}).

The points $F,G,H,J$ lie
 in the union of beams formed by the beams $DE:CB,D:AB$ and $E:AC$. Since beam $DE:CB$
 is type 2, if there exists a point in its beam region then there  exists an empty convex 5-gon.
Thus, one of the type 1 beams $D:AB$ or $E:AC$ beams has atleast 2 points, which forms an empty convex 5-gon.
\end{proof}
\begin{figure}
  \centering
   {\includegraphics[width=0.23\textwidth]{thesis12.eps}}
   \hspace{0.1\textwidth}
  \caption{(3,2) configuration}
\label{222}
\end{figure}
\begin{figure}
   \centering
   {\includegraphics[width=0.7\textwidth]{newincv1.eps}}
   \hspace{0.5\textwidth}
   \caption{Case 1 for (4,4,1) configuration}
\label{type441}
\end{figure}
\vspace*{0.2cm}
\begin{lem}
\label{lem5}
A (4,4,1) convex layer configuration has an empty convex 5-gon.
 \end{lem}
\begin{proof} Let $F,G,H,J$ be the points of $CH(P)$. Let $A,B,C,D$ be the points of the second convex layer.
 Let $K$ be the intersection of the diagonals of the convex 4-gon $ABCD$.
 Let $E$ be the point inside the convex 4-gon $ABCD$. The union of beams $E:AB,E:BC,E:CD,E:DA$ contain the points $F,G,H,J$.

Depending upon the position of point $E$ inside the $ABCD$ convex 4-gon the proof is divided into 4 cases.
\begin{figure}
   \centering
   {\includegraphics[width=0.7\textwidth]{newinc2v1.eps}}
   \hspace{0.5\textwidth}
   \caption{Case 2 for (4,4,1) configuration}
\label{type4412}
\end{figure}

\textbf{Case 1:} When ${E}$ is in the triangle $ABK$ (see $figure$  \ref{type441}).
In this case $E:CD$ beam is covered by the union of $AE:CD$ and $EB:CD$ beams. Therefore any point in the $E:CD$ beam forms an empty convex 5-gon.
If $E:CD$  beam does not have any point, one of the 3 type 1 beams $E:BC,E:AB,E:DA$ has atleast 2 points (among $F,G,H,J$) which forms an empty convex 5-gon.

\textbf{Case 2:} When $E$ is in the triangle $KCD$ (see $figure$  \ref{type4412}).
In this case $I_{1},I_{2},I_{3},I_{4},I_{5},I_{6}$ are the only feasible regions outside the $ABCD$ convex 4-gon for the placement of $F,G,H,J$ points.
The remaining outer regions are present in the $O$ region of some convex 4-gon and hence not feasible.
The $E:CD$ beam contains $I_{4},I_{5},I_{6}$ regions. Since $E:CD$ beam is type 1 beam it can have atmost 1 point. Therefore $I_{1},I_{2},I_{3}$ regions combined should have 3 points.
From the figure we can see that the union of $I_{1},I_{2},I_{3}$ regions has 3 points which forms an empty convex 5-gon with the points $A,B$.

The case where $E$ is in the triangle $KBC$ is symmetrical to the case when it is in triangle $ABK$.
The case where $E$ is in the triangle $KAD$ is symmetrical to the case when it is in triangle $KCD$.

\end{proof}
Let ${E,F,G,H}$ be the points of the first convex layer and  ${A,B,C,D}$  be the points of the second convex layer of configuration 8.
Without loss of generality, let us assume that rays $\overrightarrow{AB},\overrightarrow{DC}$ intersect and rays
 $\overrightarrow{DA},\overrightarrow{CB}$ intersect.

 \begin{figure}
   \centering
   {\includegraphics[width=0.6\textwidth]{bad44case1v1.eps}}
   \hspace{0.5\textwidth}
   \caption{When no pair of rays intersect}
\label{bad44case1}
 \end{figure}

\begin{lem}
\label{lem6}
The 4 type 2 beams AB : FE,BC : GF,DC : GH,AD : HE  cover the entire outer region in configuration 8.
\end{lem}
\begin{proof}
We consider 2 cases depending on the intersections of the rays $\overrightarrow{AE},\overrightarrow{BF},\overrightarrow{CG},\overrightarrow{DH}$.

\textbf{Case1:} When none of the rays  $\overrightarrow{AE},\overrightarrow{BF},\overrightarrow{CG},\overrightarrow{DH}$ intersect each other (see $figure$ \ref{bad44case1}).

The  regions  covered by $AB : FE,BC : GF,DC : GH,AD : HE$ beams are correspondingly $O_{1},O_{2},O_{3},O_{4}$ beam regions
which cover the entire outer region in configuration 8.

\textbf{Case2:}  When some of the rays  $\overrightarrow{AE},\overrightarrow{BF},\overrightarrow{CG},\overrightarrow{DH}$ intersect each other (see $figure$ \ref{bad44case2}).

Now we prove that atmost one of the pairs of rays $(\overrightarrow{AE},\overrightarrow{BF})$ or $(\overrightarrow{CG},\overrightarrow{BF})$
 can intersect in configuration 8. Assume that $(\overrightarrow{AE},\overrightarrow{BF})$ intersect (see $figure$  \ref{bad44case2}).
The case $(\overrightarrow{CG},\overrightarrow{BF})$ intersecting is symmetrical.

 Consider $EA$ and $FB$, extend them inside $ABCD$ convex 4-gon until they intersect with $CD$.
Let the points of intersection be ${J,K}$. When $GC$ is extended inside the $ABCD$ convex 4-gon it has to intersect
$BK$ before it intersects $AB$ or $AD$. Hence $(\overrightarrow{BF} , \overrightarrow{CG})$ do not
intersect. By a similar reasoning, ray $\overrightarrow{HD}$ intersects $AJ$ before it intersects $AB$ or $BC$.
Hence $(\overrightarrow{AE} , \overrightarrow{DH})$ do not intersect.
 The pair of rays $(\overrightarrow{CG} , \overrightarrow{DH})$ do not intersect because the pair of rays $(\overrightarrow{BC} , \overrightarrow{AD})$
do not intersect.

The  regions  covered by $AB : FE,BC : GF,DC : GH,AD : HE$ beams are correspondingly $O_{1},O_{2},O_{3},O_{4}$ beam regions (see $figure$  \ref{bad44case2})
which cover the entire outer region in configuration 8.

\end{proof}
 \begin{figure}
   \centering
   {\includegraphics[width=0.6\textwidth]{bad44cas2v1.eps}}
   \hspace{0.5\textwidth}
   \caption{When one pair of rays intersect}
\label{bad44case2}
\end{figure}
\begin{figure}
  \centering
   {\includegraphics[width=0.6\textwidth]{prconv1.eps}}
   \hspace{0.1\textwidth}
   \caption{Configuration 8}
\label{y}
\end{figure}
\begin{lem}\label{proof for config8}
Any point added to configuration 8 forms an empty convex 5-gon.
\end{lem}
\begin{proof}

We consider 2 cases depending on whether the point is added inside $EFGH$ convex 4-gon or outside $EFGH$ convex 4-gon (see $figure$ \ref{y}).
\vspace{1cm}

\textbf{Case1:} Placing the point inside $EFGH$ convex 4-gon:

The point added lies in $I$ region,
$S$ region, $Z$ region or  $O$ region of $ABCD$ convex 4-gon.
 If the point is placed in the $O$ region of $ABCD$ convex 4-gon then there exists an empty convex 5-gon.
 If point is placed  in the $S$ region of $ABCD$ convex 4-gon then it forms a (4,3,2) convex layer configuration which
contains an empty convex 5-gon (lemma \ref{lem4}).
 If the point is placed in the $I$ or $Z$ region of the $ABCD$ convex 4-gon then it forms a (4,4,1) convex layer configuration which
contains an empty convex 5-gon (lemma \ref{lem5}).

\textbf{Case2:} Placing the point outside the $EFGH$ convex 4-gon:

By lemma \ref{lem6}, the 4 beams $AB:FE,BC:GF,DC:GH,AD:HE$  cover the entire outer region.
The added point lies in one of the 4 type 2 beams and forms an empty convex 5-gon.
\end{proof}

\begin{thm}
 The empty convex 5-gon game always ends in the 9th step.
\end{thm}
\begin{proof} The game does not end in the 5th step because no 4 point sets  are bad configurations. By lemma \ref{lem1}, the game will reach
either configuration 6.1 or 6.2. Since all 6 point sets are not bad configurations (lemma \ref{added}), the game will not end in the 7th step.
 By lemma \ref{lem2} the game will reach either configuration 7.1 or 7.2. By lemma \ref{le3}, the game will reach configuration 8 and finally
from lemma \ref{proof for config8}
the game ends in the 9th step and
player 2 wins the game.
\end{proof}

\section*{Conclusion}
In our paper we have introduced the two player game variant of Erd\H{o}s-Szekeres problem
and proved that the game ends in the 9th step for the convex 5-gon and empty convex 5-gon game and player 2 wins in both the cases.

One natural question would be to analyze the game for higher values of $k$ i.e. determine $N_{G}(k)$ and $H_{G}(k)$ for $k > 5$.
Our approach will be very tedious for higher values of $k$ as with the increase in the number
of points in the point set, the number of point configurations increases exponentially.

We have shown that configuration 8 is a bad configuration for the empty convex 5-gon game.
Another natural question is to determine whether there exists bad configurations for $k > 5$.
 More specifically, does there exist point configurations with the property that any point added to this configuration
 forms an empty convex $k$-gon or convex $k$-gon for $k > 5$. A negative result for the above question gives a lower bound for $N_{G}(k)$ and $H_{G}(k)$.

redpajama_set_name: RedPajamaArXiv


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What to Expect this Labor Day Weekend in Zion
Labor Day weekend is upon us. It is always one of the busiest weekends of the year at Zion. With the anticipation of a busy weekend, park officials are asking those who are planning to visit the park this weekend, to plan ahead, especially with…
https://www.zioncanyon.com/wp-content/uploads/2020/09/Zion-Shuttle-2019-1484x1113-1.jpg 1113 1484 olwm https://www.zioncanyon.com/wp-content/uploads/2015/02/zion-canyon-logo-white-fnt.png olwm2020-09-02 19:55:002020-09-02 19:55:00What to Expect this Labor Day Weekend in Zion
Activities, Activities in Zion, Aerial, ATV, Bed and Breakfast, Biking, Biking in Zion, Campground, Camping, Canyoneering, Canyoneering in Zion, Concerts, Dining, Entertainment, Fishing, Hikes Near Zion National Park, Hiking, Hiking in Zion National Park, Horseback Riding, Hotels, Lodging, Movies, Parks Near Zion, Restaurants, RV Park, Shopping, Springdale, Things To Do In Zion, Tours, Vacation Rentals, Zion Park, Zion Park Hikes
Zion National Park Week
https://www.zioncanyon.com/wp-content/uploads/2019/04/Zion-National-Park-Week.jpg 852 1280 olwm https://www.zioncanyon.com/wp-content/uploads/2015/02/zion-canyon-logo-white-fnt.png olwm2019-04-23 17:46:232019-04-24 15:37:14Zion National Park Week
Explore the Pleasant Waterfall at Lower Pine Creek
https://www.zioncanyon.com/wp-content/uploads/2019/04/Lower-Pine-Creek.jpg 1125 1500 Alexis Barrett https://www.zioncanyon.com/wp-content/uploads/2015/02/zion-canyon-logo-white-fnt.png Alexis Barrett2019-04-02 18:55:332019-04-08 17:29:49Explore the Pleasant Waterfall at Lower Pine Creek
Explore Observation Point Trail in ZNP
https://www.zioncanyon.com/wp-content/uploads/2019/04/Observation-Point-slider.jpg 500 2000 Alexis Barrett https://www.zioncanyon.com/wp-content/uploads/2015/02/zion-canyon-logo-white-fnt.png Alexis Barrett2019-04-02 18:35:232019-04-08 17:31:18Explore Observation Point Trail in ZNP
Activities, Activities in Zion, Campground, Camping, Hikes Near Zion National Park, Lodging, Outdoor, Parks Near Zion, RV Park, Springdale, Things To Do In Zion, Visitor Center, Zion Park
Zion Campgrounds Now Requiring Reservations
Recently, a popular website boasted that Zion National Park is the most beautiful place on earth. Of course, how do you compare the white-sand beaches of Amanpulo with the mountainous regions of the Southwest? How do you compare a five-star…
https://www.zioncanyon.com/wp-content/uploads/2018/02/25137089973_ec69834b69_k.jpg 1536 2048 olwm https://www.zioncanyon.com/wp-content/uploads/2015/02/zion-canyon-logo-white-fnt.png olwm2018-02-28 23:51:122018-06-02 19:30:57Zion Campgrounds Now Requiring Reservations
Activities, Activities in Zion, Bed and Breakfast, Biking, Concerts, Dining, Entertainment, Hotels, Lodging, Outdoor, Restaurants, Shopping, Springdale, Theaters, Things To Do In Zion, Visitor Center, Zion Park
Downtown Springdale
Springdale is well-known for being the "Gateway to Zion." While it is true that Springdale hosts travelers who are visiting to explore Zion National Park, there are so many things to do in downtown Springdale that it would be a shame to…
https://www.zioncanyon.com/wp-content/uploads/2018/02/3760162311_fd3ed0c1bc_b.jpg 641 1024 olwm https://www.zioncanyon.com/wp-content/uploads/2015/02/zion-canyon-logo-white-fnt.png olwm2018-02-23 20:43:252018-03-10 01:24:55Downtown Springdale
Activities, Bed and Breakfast, Entertainment, Hotels, Lodging, Outdoor, Restaurants, Things To Do In Zion, Vacation Rentals, Zion Park
Arrowhead Bed and Breakfast
You are now in the heart of the Old West. Arrowhead lies just off Highway 89 otherwise known as National Park Highway or the Backbone of the Rockies. Your experience at Mt. Carmel Junction, Utah will be one of the great adventures upon which…
https://www.zioncanyon.com/wp-content/uploads/2018/01/Ah-Piney-front-8-13-840x630.jpg 630 840 olwm https://www.zioncanyon.com/wp-content/uploads/2015/02/zion-canyon-logo-white-fnt.png olwm2018-01-19 18:36:132020-05-19 19:55:41Arrowhead Bed and Breakfast
Activities, Activities in Zion, Lodging, Zion Park
Winter Information For Zion – Zion Ponderosa
https://www.zioncanyon.com/wp-content/uploads/2015/02/zion-canyon-logo-white-fnt.png 0 0 Tailer Holt https://www.zioncanyon.com/wp-content/uploads/2015/02/zion-canyon-logo-white-fnt.png Tailer Holt2016-11-03 19:50:292016-11-03 19:50:29Winter Information For Zion – Zion Ponderosa
Activities, Biking, Biking in Zion, Lodging, Outdoor, Vacation Rentals
This. Is. Yurt. Life.
https://www.zioncanyon.com/wp-content/uploads/2016/04/Goose11.jpg 685 913 Hollie Reina https://www.zioncanyon.com/wp-content/uploads/2015/02/zion-canyon-logo-white-fnt.png Hollie Reina2016-04-20 18:17:542018-05-02 16:53:14This. Is. Yurt. Life.
Activities, Activities in Zion, Hiking in Zion National Park, Lodging, Things To Do In Zion
One of the Best Sunset Lookouts in Zion
https://www.zioncanyon.com/wp-content/uploads/2015/02/zion-canyon-logo-white-fnt.png 0 0 Clay Hunter https://www.zioncanyon.com/wp-content/uploads/2015/02/zion-canyon-logo-white-fnt.png Clay Hunter2016-03-02 18:38:002016-03-02 18:52:02One of the Best Sunset Lookouts in Zion
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It is a touching story from Craig Schwab of Glendale, NY. Touching not solely as a result of Craig writes frankly of a pal who's fading away, however as a result of it's a story that increasingly more child boomers might inform as we speak. He calls it, The Lacking Piece.
I take into consideration how simply pals turned pals after we had video games to play. There was a simplicity to picking sides on a ball subject or a basketball court docket. Sizing each other up for successful a sport later. At first, all anybody needed to do was play.
That's how I met my greatest pal in 1968. We have been each 12 years outdated. His father was the one who launched us. I used to be enjoying basketball with different guys from the neighborhood. It's amusing to me now, I knew all of these different guys earlier than assembly my buddy Tom. I don't keep in mind any of their names now.
That's how our friendship started. Tom and I have been inseparable as youngsters. He was in my marriage ceremony celebration after I tied the knot in 1978. He watched me turn out to be a father and grandfather. I watched him expertise the identical life-affirming accomplishments.
Tom Huber and Craig Schwab at Louis Armstrong Museum.
Within the ebook I inform tales that solely he and I might totally admire. Tales from our shared 50 years collectively as pals. Our many journeys to ebook shops everywhere in the East Coast. I inform him within the ebook to cease playing around. I desperately wish to imagine he's nonetheless able to pulling my leg.
The ebook, I now notice, was actually for me. In it, I attempted to piece collectively all the things I might to carry him again. However the lacking piece is, I can't flip again time. I can't reverse what's already fading away.
I present Tom photos on my telephone. Each now and again he will get a glance of recognition on his face. I'm going residence pleased, pondering Tom had an excellent day. We had an excellent day.


text:
Miro Antic has been elected one of The 100 Most Influential People in the Event Industry
Mr. Miro Antic, Co-founder and Chief Experience Officer at marketing agency M2Communications has been elected one of The 100 Most Influential People in the Event Industry, by Eventex.
I want to thank you all, who voted for me and launched me among 100 Most Influential People in Event Industry. It is up to the M2C team and me now to set new creative goals, which will bring innovative solutions in the field of event organization and at the same time, be an inspiration to others - Miro Antić said.
The vote was public, on the internet, and 476 event industry professionals were nominated by the public for a chance to make it to the top 100. Final list was created via public vote, as well.
Miro has over 15 years of international experience in marketing and business development.
He is an expert in launching of premium brands and their positioning in highly competitive environments and global market as well as creator of multimedia projects and spectacular events.
Eventex Awards was founded in 2009. to celebrate creativity, innovation and effectiveness in the industry.
M2Communications is one of the leading event agencies in the region. M2C is a multiple award - winning agency.
Eventex Top 100
Official Press Release