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Home/Brewery Spotlight/Cycle Brewing Company
Cycle Brewing Company
Staff Account May 6, 2020
This much sought-after brewery in Florida has been operational since 2012. Reviews of the place on popular beer sites such as Untappd and BeerAdvocate show that, despite being in the businesses for a while now, the brewery shows no sign of slowing down and still remains as popular as it has been since its opening day.
About Cycle Brewing
The brewery, located in a space of modest dimensions, is founded by Doug Dozark, who is also the head brewer. A lover of small batches of fresh beer and barrel ales aged for long periods of time, Doug has always been keen on sharing his passion with people while continuing to experiment with the craft at the same time. Initially, Cycle Brewing used to sell their aged and non-aged drinks in crowlers, bottles, and straight from the tap. But since 2018, they have also included cans to their stock. Cycle Brewing is undoubtedly Doug's dream project, but also a dream shared by his family. He is supported and partnered in business by his wife, Tara, along with a few other members of his close family. The tight-knit crew believes in serving some of the best beers and running a business that helps them achieve the lifestyle that they want. Rather than going by statistics, peer pressure, and other such factors, Cycle prefers to quietly focus on improving and expanding their beer-making techniques, especially the fermentation and barrel-ageing process.
For example, in 2016, Cycle Brewing was one of just the seven breweries in the country that used Jameson Irish Whiskey barrels to age some of their finest beers and produce a malty, sweet drink in collaboration with Caskmate Drinking Buddies.
Image by Bill Barlow on Google Maps
Growing Notoriety
In a world of rapidly-growing brewery establishments, Doug's policy is that, rather than following the pack, they would stick to their winning methods and keep creating drinks that have been popular for the past several years. He admits to a few twinges of jealousy occasionally, but when they really sit down and think about it, they find that their model of business and making beer are what their customers have loved for so many years. Doug prefers to keep that tradition alive and keep the business well-managed within the family itself. Doug does plan on increasing his production capacity and supply beer to a larger number of retailers, but the focus is and always has been on finding the choicest beers to fill their crowlers and cans. The brewery is determined to innovate their brewing process without compromising their core drinks – mainly the dark beers that they have been so famous for all these years.
Cycle Brewing has received recognition and awards at several events and surveys. The most notable of these is when it was adjudged to one of the top ten breweries in the world by reputed breweries' internet hub RateBeer, in 2018.
Until 2018, the brewery used to primarily sell its beers at its premises. But when they signed a contract with Brew Hub, they increased the volume of their business substantially, as they now have a distributor who could get their beers across the entire state. What's more, customers who could not make it to Cycle's taproom can now taste some of their delicious drinks from packaged cans. Cycle had started off by selling two of its most popular drinks – Cream & Sugar, Please and Crank IPA – in cans. Now, customers can also enjoy Fixie, Download, Shallow Flake, and Name Goes Here in cans, which are available in a few places outside Florida too, in addition to being available abundantly within the state.
Image by Brian Smith on Google Maps
What Do Customers Have to Say?
Located in St. Petersburg, Florida, more about Cycle brewing Company can be learned from their social media acounts on Instagram and Facebook, along with their website. But before you start scouring the internet for Cycle Brewing's special beer cans, let's have a look at what people have had to say about the place and its drinks so far. The first thing that people have loved about the place is its vibe. Rather than being a typical family-friendly place, Cycle Brewing is known more for its ultra-cool atmosphere that'll make you feel like sipping on their IPAs in your most laid-back clothes. It's just your beer, you, and your relaxed senses.
The brewery permits its customers to carry their own food and also well-trained pets. Sours, stouts, and any other beer that falls in the dark brew category seem to be the most popular drinks here, with people who don't usually prefer stouts also thoroughly enjoying a glass of it here. The lack of food at the brewery is not a deterrent at all because there are plenty of restaurants that serve lip-smacking food a stone's throw away from Cycle Brewing. Another interesting feature of the place that the customers seem to really dig is the table – it's unique because it's really just a big rock, or rather a mini-boulder, that's been shaped into a large table that people can sit around and enjoy their brews. The place also features some interesting motorbike parts in their interior decoration, thus making it a treat for virtually all categories people. The doors of Cycle Brewing are open to all!
You can visit the Cycle Brewing Company at 534, Central Avenue, St. Petersburg, Florida 33701. To know more about this brewery, you can also visit their official website.
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PLEASE NOTE: Due to high volume of orders, shipping may be delayed for orders placed between Thursday, March 21 and Sunday, March 24. We are working to catch up on these orders and you will receive email confirmation for shipping updates when we have them.
*Orders must be placed before noon to be guaranteed 2-day or next-day delivery.
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-Orders placed over the weekend or on holidays will ship the next available business day.
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-NOTE: This service may not be available for certain addresses and select items like fuel and oversized items like luggage or boats.
-Kayaks and SUPs are only available with Kayak/SUP shipping, which is a $250 fee. We will calculate the specific shipping fee once the order is placed and if it is less than $250, we will modify the fee on your order based on the actual cost.
-Uncle Dan's is not responsible for stolen or missing packages that have a clear proof of delivery, however, we take each situation on a case by case basis and will do our best to help you in whatever way we can!


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En teoría de grafos, una rama de las matemáticas, muchas familias importantes de grafos se pueden describir mediante un conjunto finito de grafos individuales que no pertenecen a la familia y además excluyen todos los grafos de la familia que contienen cualquiera de estos grafos prohibidos como subgrafos o menores (inducidos).

Un ejemplo prototípico de este fenómeno es el teorema de Kuratowski, que establece que un grafo es plano (es decir, se puede dibujar sin cruces en el plano) si y solo si no contiene ninguno de los dos grafos prohibidos, el grafo completo  y el grafo bipartito completo . Para el teorema de Kuratowski, la noción de contener es la del homeomorfismo de grafos, en la que una subdivisión de un grafo aparece como subgrafo del otro. Así, todo grafo o bien tiene un dibujo plano (en cuyo caso pertenece a la familia de los grafos planos) o tiene una subdivisión de al menos uno de estos dos grafos como subgrafo (en cuyo caso no pertenece a los grafos planos).

Definición
De manera más general, una caracterización de grafos prohibidos es un método de identificar una familia de estructuras de grafos o hipergrafos, mediante la especificación de subestructuras cuya existencia está prohibida dentro de cualquier grafo de la familia. Las diferentes familias varían en la naturaleza de lo que está prohibido. En general, una estructura G es miembro de una familia  si y solo si una subestructura prohibida no está contenida en G. Las subestructuras prohibidas podrían ser:

 Subgrafos, grafos más pequeños obtenidos a partir de subconjuntos de vértices y aristas de un grafo más grande.
 Subgrafos inducidos, grafos más pequeños obtenidos seleccionando un subconjunto de los vértices y usando todas las aristas con ambos extremos en ese subconjunto.
 Subgrafos homeomorfos (también llamados menores), grafos más pequeños obtenidos a partir de subgrafos al colapsar caminos de vértices de grado dos en aristas simples.
 Menores, grafos más pequeños obtenidos de subgrafos por contracción de aristas arbitrarias.

El conjunto de estructuras que tienen prohibido pertenecer a una familia de grafos dada también puede denominarse conjunto de obstrucciones para esa familia.

Las caracterizaciones de grafos prohibidos se pueden usar en algoritmia para probar si un grafo pertenece a una familia determinada. En muchos casos, es posible probar en tiempo polinómico si un grafo dado contiene alguno de los miembros del conjunto de obstrucciones y, por lo tanto, si pertenece a la familia definida por este conjunto.

Para que una familia tenga una caracterización gráfica prohibida, con un tipo particular de subestructura, la familia debe estar cerrada bajo subestructuras, es decir, cada subestructura (de un tipo dado) de un grafo de la familia debe ser otro grafo de la familia. De manera equivalente, si un grafo no es parte de la familia, todos los grafos más grandes que lo contengan como subestructura también deben excluirse de la familia. Cuando esto es cierto, siempre existe un conjunto de obstrucción (el conjunto de grafos que no están en la familia pero cuyas subestructuras más pequeñas pertenecen todas a la familia). Sin embargo, para algunas nociones de lo que es una subestructura, este conjunto de obstrucciones podría ser infinito. El teorema de Robertson-Seymour prueba que, para el caso particular de los menores, una familia cerrada por menores siempre tiene un conjunto finito de obstrucciones.

Lista de caracterizaciones prohibidas para grafos e hipergrafos

Véase también
Conjetura de Erdős-Hajnal
Problema del subgrafo prohibido
Menor matroide
Problema de Zarankiewicz

Referencias

Teoría de grafos
Menor en teoría de grafos
Familias de grafos
Teoría de hipergrafos

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Q: Parameterized REST @Path I am at loss on how to get the parameterized @PATH to work.
Here is my web.xml
<servlet-mapping>
    <servlet-name>JerseyServlet</servlet-name>
    <url-pattern>/ND/*</url-pattern>
</servlet-mapping>

Here is my resource class:
@Path("/ND")
public class TransactionResource
{
@Context UriInfo uriInfo;

public TransactionResource()
{
}

@GET
@Produces(MediaType.TEXT_PLAIN)
public String itWorks()
{
    return String.format("Get is OK. %s", DateUtil.now());
}

@GET @Path("/NJ")
@Produces(MediaType.TEXT_PLAIN)
public String itWorksForState()
{
    return String.format("Get is OK for NJ. %s", DateUtil.now());
}

@POST
@Produces(MediaType.TEXT_PLAIN)
@Consumes(MediaType.APPLICATION_XML)
public String addTransaction(Transaction pTransaction) throws Exception
{
    //some code here
    return "Successful Transmission";
}

When I do a GET or POST at the URL http://my_web_app:8080/ND then both methods work fine.
But for some reasons, the GET method at URL http://my_web_app:8080/ND/NJ always return the 404-NotFound.
What have I done wrong here?
Thanks

A: You have 4 level of path :

*

*The path of your web application Context in the server : Maybe myapp

*The path of the Jax-Rs servlet in web.xml : Here /ND/, but I would suggest /ws

*The path of the Resource : The first @Path above the class. You should probably have @Path("transaction")

*Then an optional @Path above each method. Let's say that you dont add any @Path in any method.

Now you have
    @Path("transaction")
    public class TransactionResource
    {
    @GET
    @Produces(MediaType.TEXT_PLAIN)
    public String itWorksForState()
    {
      return String.format("Get is OK for REST. %s", DateUtil.now());
    }
}

Go to Firefox and type
 http://my_web_app:8080/myapp/ws/transaction : you should read the Date
If ever you add
    @Path("morepath")
    @GET
    @Produces(MediaType.TEXT_PLAIN)
    public String itWorksForState()
    {
      return String.format("Get is OK for REST. %s", DateUtil.now());
    }

Then you must go to http://my_web_app:8080/myapp/ws/transaction/morepath

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Welcome to the Old Church Dental Practice In Halifax.
Located just 9 miles outside Leeds City Centre, and a mere 3 miles from Halifax, you'll find the charming Yorkshire village of Lightcliffe, home to this state-of-the-art dental and implant clinic, quite unlike any you've visited before.
Our exclusive practice is housed in a Grade I listed church building with beautiful original stone-work, stained glass windows and antique furnishings. Our experienced team take great pride in making you feel relaxed and comfortable, no matter how nervous you may be.
At the Old Church we cater for everyone, with treatments ranging from simple dental examinations, to complex full mouth rehabilitation, using the latest scanning and implant technology. Finance options are also available to make your treatment more affordable.
Why not call us today and see how we can help make you smile?
One of our most sought after treatments is denture stabilisation, or denture replacement, for people wishing to be rid of troublesome, ill-fitting dentures. Doctor Nigel Hargreaves, our resident Implant Surgeon at The Old Church, regularly replaces dentures with full-arch bridges on implants. This is the closest thing to getting your natural teeth back. The procedure can be completed in one day, and you can eat on your new teeth immediately.
Nigel qualified from Sheffield in 1988 with the KaVo prize in Oral Surgery. He went on to do a Masters Degree in Restorative Dentistry. He has worked in private practice for over twenty years and completed advanced training in London, Austria, Italy, Portugal and the USA in implant, cosmetic and restorative dentistry alongside some of the world's leading clinicians.
After working in Manchester, Nigel returned to Yorkshire and built two successful implant referral practices. Nigel's passion is smile and functional reconstructions, whether using veneers, crowns, natural teeth or integrating dental implants where teeth are missing.
We often hear that having dentures, or missing teeth, means that you struggle to eat certain foods, lack the confidence to smile, or are reluctant to socialise for fear of being judged by the appearance of your teeth.
We can solve many of these issues by restoring your smile using dental implants.
The benefits of dental implants has been proven time and again as the best way of replacing missing teeth, or getting rid of your dentures. Dental implants can be used to replace one missing tooth, or an entire jaw of missing teeth and they will look, feel and behave just like your natural teeth. Implants also work to preserve the natural bone structure and tone of the facial muscles to avoid wrinkling and premature ageing.
No one is born with a perfect smile. However, everyone can benefit from QuickStraightTeeth™, one of the the quickest and gentlest ways to straighten teeth. All you need to achieve a movie star smile, without the movie star price tag.
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We offer a range of cosmetic treatments to include veneers, tooth whitening, cosmetic bonding and the treatment of gummy smiles.
Veneers are extremely versatile, and can be used to quickly disguise a number of imperfections, which is why they are often used for smile makeovers. Tooth whitening is another popular cosmetic treatment. We offer a tooth whitening treatment you can do in the comfort of your own home. Cosmetic bonding can be used in many situations, such as repairing chipped teeth, improving the appearance of discoloured teeth, close gaps or even change the shape of the tooth. Some people feel self-conscious because of the excessive amounts of gum they show when smiling. This is known as a gummy smile and can be treated with gum contouring.
One solution for missing teeth is Cosmetic Dentures. This is a relatively cost-effective solution to the problem of eventual tooth loss and should be considered as a viable option if this problem starts to affect your life. We understand the impact that losing teeth can have on the quality of life and we know it can affect your confidence, alter your facial appearance and interfere with eating.
At The Old Church, we routinely use dental implants to secure loose dentures, and we can even replace your dentures completely with a full mouth of teeth, fixed in place that look, behave and function just like your natural teeth – even you would struggle to know they were not your own teeth! This allows you to regain a beautiful smile and eat whatever you wish.


Good and the Celebrity Culture.
Notice the clarity and insight of these words. If you try to understand the "prosperity" of the wicked without an eternal perspective, it will be oppressive. This is a vital truth that you must pass on to your children. The world paints a deceptive picture of disobedience. The allure of the world will lose its attraction as it is displayed in the light of eternity.
The stark reality of this psalm serves as a warning to your children of how attractive the world can appear. That is why it is essential that your parenting be based on one solid truth–the nearness of God is your good! The world will appeal to your children with the same force with which it appealed to the psalmist. God's goodness must be real to you, so real that your life is dominated by it. The nearness of God–is it your good?
The world of the celebrity has dominated the news. Some have fallen. Others rise to take their turn. They, too, will appear to have it all together. Pray that God will give you joy in being near to him. Show your children how much God's nearness means to you. Don't mislead them by allowing other things to crowd between you and your God. If your children are struggling with various things, delight in them by working with them to value the nearness of God. The worth of your children is not determined by things that can be measured by the world's standards. Don't allow the performance of your children to become the barometer of what is good for you.
Remember, you do not want to take your children to where you are. You want to take your children to the same place you are longing for–the cross of Christ.


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By James F. Kutch Funeral Home - Crematory | Posted in: James F. Kutch Funeral Home - Crematory
Alva B. Vasilich, North Versailles, passed away Sunday, January 10, 2021. She was the daughter of the late Joseph and Millie (Mauger) Bertozzi. Loving wife of 66 years of the late Thomas E. "Babe" Vasilich. Beloved mother of James J., of Las Vegas, Nevada, Thomas E. Jr. (Mary Jane) of Port Vue. Grandmother of Stacey (Chuck) Hughes, Thomas J., & Matthew J. Vasilich and the late Kaitlin Vasilich. Great grandmother of Trevor and Brielle Hughes.
¶ | James F. Kutch Funeral Home - Crematory
On Sunday, January 10, 2021, Joseph "Papa Joe" Juricich, 96 of North Versailles passed away. Son of the late Joseph & Katherine (Stemberger) Juricich. Beloved husband of the late Constance Juricich. Loving father of Karen (Don) Elko, Joey P. (Barb) Juricich, Michael Juricich, Pamela (Doug) McGinnis, and the late Jeffrey Juricich. Brother of the late Irene Rosa. Dear Uncle of Rich (Ginny) Rosa, Kathy (Jan) Fung, and Theresa Nuzzo. 10 grandchildren and 10 great grandchildren. Stepfather to Barb (Jim) Romanovich and Butch (Karen) Wilson. Joe and his parents were lifetime members of SNPJ. He was a member of the Bricklayers Union local 9 and a 55-year member of the Norwin Elks. Joe was a property developer and home builder, along with his father, Joe, and his son, Joe, (J&J Builders). Joe's life continued to be one victory after another.
Joanna Marie (Waszkiewicz) Detwiler
Joanna Marie (Waszkiewicz) Detwiler, age 66, of East McKeesport, on Thursday December 31, 2020, passed away surrounded by her family. Beloved wife of Dean K. Detwiler. Loving Mother of Charles (Kelli) and John (Karry) Detwiler. Grandmother to Haylea (Joseph) Curtin, Nathanael, and Nicholas. Great-Grandmother to Grayson and Easton. Joanna is predeceased by her brother John (Jerilynn) and Raymond (Marlene) Waszkiewicz. She is survived by her brothers Francis "Jerry" (Lori), Eugene (Marilyn) and Edward (Nancy) Waszkiewicz. Also, survived by numerous nieces and nephews.
Dennis Paul Bernhardy
Dennis Paul Bernhardy, 81, of East McKeesport died Tuesday, December 15, 2020 at his home.
Dennis was born June 10, 1939 in East McKeesport, PA. He graduated in 1957 from East McKeesport High School. He then went on to follow in his father's footsteps of carpentry and employment with the National Tube Works Mill in McKeesport.
Dennis was actively involved in his community, a devoted Christian and member of Linway United Presbyterian Church, a member of the Freemasons, delivered for Meals-On-Wheels for many years, decades-long member with multiple volunteer fire departments, a steadfast leader in multiple youth sport programs, and most importantly, a dedicated father and husband.
Frances Margaret Antico
Frances Margaret Antico, 76, of Oakmont, formerly of Glassport, died Friday December 18, 2020. Beloved Daughter of the late Frank Antico and Amelia (Reno) Antico, she is survived by her brother, Dominic Antico of Murrysville and sister, Veronica Zalewski of Park Forest, Illinois.
Frances graduated from Glassport High School and was a life-time resident of Glassport, Pennsylvania.
John "Jack" Liebert
John "Jack" Liebert, 69, of McKeesport, died Nov. 27, 2020 after a long and courageous battle with cancer. Son of the late William and Jennie (Fontanello) Liebert. Beloved husband of Mary (Chismar) Liebert. Loving father of Chad (Stacie) Liebert, Jason Liebert, Angela (Dan) Limbacher, Tim Sweeney, and Erin (Ashley) Merida. Dear grandfather of Brandon, Soloman, Jewel, Grace, Ethan, and Carter. Jack enjoyed golfing, his motorcycle, the outdoors, and time spent with his family.
Dennis "Denny" Pascuzzi
Dennis "Denny" Pascuzzi, 78, died Nov. 11, 2020. Son of the late William and Gloria Pascuzzi Brother of the late Douglas Pascuzzi and David W. Altman, Sr. Dennis is survived by his nephew: David W. Altman, Jr, and his wife, Heidi and their two children, Brittany and Blake, and also good friend Shawn Conley. Dennis's passions included golf and cutting hair. Both were practiced for 50 years. Dennis was the owner of The Hair Den where he was loved by all his clients. Dennis will be remembered by his love for Pittsburgh sports, fishing, cars, but most importantly his love for others and his family.
George W. Lokay
George W. Lokay, 74, of North Versailles Twp. died Thursday, November 12, 2020, after a long and courageous battle with cancer. Son of the late George and Lucile (Lacivita) Lokay. Beloved husband of Gloria (Iaci) Lokay. Loving father of Gina (Mike) Giacchino and George (Rachel) Lokay. Brother of Jim (Angie) Lokay, Jack (Renee) Lokay, Rich (Debbie) Lokay, and Tim (Judy) Lokay. Dear grandfather of Nicoletta (Travis) Kuhn, Stefano Giacchino, Francesca Giacchino, and Sydney Lokay. Great Grandfather of Gia Kuhn.
George was a Police Officer in North Versailles for 30 years. He enjoyed golfing, bowling, the outdoors, working in his garden and time spent with his grandchildren.
Kevin D. Flanagan
Kevin D. Flanagan, 70, of North Versailles Twp., formerly of Meadville, PA and Baltimore, MD died Friday October 30, 2020. Son of the late Carroll and Adelaide (Fontz) Flanagan. Beloved husband of Vicki (Konopka) Flanagan.
Loving father of Robert Love, Mathew (Linda) Love, Sean (Kate) Flanagan, Ryan Flanagan, and Jamie Flanagan Hession. Brother of Terence (Barb) Flanagan and Kerry (Joe) Flanagan Upman.
Frank Tigano
Frank Tigano, 95, died peacefully on Sunday, October 18, 2020. He was born in Braddock, Pennsylvania on March 2, 1925, the son of the late Joseph Tigano and Rose (DeMarco) Tigano Canalungo. He is survived by his wife of 71 years Clara (Zezzo) Tigano, and four children, Frank (Madelain) Tigano, David (Fran) Tigano, Rosalie (Stephen) Carpenter, and Jerry (Jennifer) Tigano. He has 7 grandchildren, 15 great grandchildren and 6 great, great grandchildren. Frank was a U.S. Army veteran of World War II. He was the founder and co-owner of F & C Distributing Company in Braddock and Irwin.
Ray Leonard Cornell
Ray Leonard Cornell, 82, of White Oak died Friday, October 16, 2020. Beloved husband of Laura M. (Fagley) Cornell. Loving father of Michael (Serena) Cornell , Richard (Leanne) Cornell, and Elizabeth (Rick) Huhn. Dear grandfather of Calvin, Ariel, Sarah, Trent, and Catherine. Brother of Patricia Scanlon and uncle of Roger and Ryan Scanlon. Close cousin of Jean Hedderick, Kate Cornell, and Rob Cornell.
Lawrence L. Curran
Lawrence L. Curran, 73, of North Huntingdon Twp. died Sunday, Sept. 27, 2020. Larry is survived by his beloved wife of 49 years, Margery A (Timko) Curran; daughter, Valery L McCann; son, Ryan D Curran (Rita); grandsons, William, Colin, and Trevor; and brother Charles R Curran Jr.
He was preceded in death by his father, Charles; mother, Irene; sister, Patricia Galata. He was born Oct. 5, 1946 in Homestead, PA and was a 1964 graduate of West Mifflin North High School. He was a veteran of the US Army serving in Vietnam where he received a Purple Heart.
Larry worked for United States Steel in Homestead for 21 years and later as a Letter Carrier in McKeesport for 24 years. He served as Executive Vice President of the NALC Branch 332.
Georgine J. "Georgia" Wagner
Georgine "Georgia" Wagner, age 95, passed away peacefully on September 22, 2020 at home.
Born in Underodoch, Germany on July 10, 1925, she was the daughter of the late Max and Sophie (Kestle) Mueller, and for 65 years the wife of the late Edward Wagner. Georgia was also preceded in death by her brother Hank Mueller. She is survived by her children, Ed (Jeannie) Wagner, and Sherry (the late John) Scharritter; five grandchildren, Gary (Kissy), Kelly, Scott (Janine), Erik, Chris (Loree), 8 great grandchildren, and several nieces and nephews. Georgia took great pride in caring for her family and home.
She was a member of the former Immanuel Lutheran Church in Braddock.
Virginia A. Piper-Fitzroy
Virginia A. Piper-Fitzroy, 70, of East McKeesport died Sunday, August 30, 2020. Daughter of the late Gerald and Rebecca Piper. Beloved wife of Robert Fitzroy. Loving mother of Dwyer ( Beth Ann) Lynch, Gerald (Nicole) Lynch, David (Diana) Lynch, and Jake (Melanie) Fitzroy. Dear Grandmother of Harmon Lynch, Jamisen Fitzroy, Avery Fitzroy, Logan Fitzroy, and Nicolas Lynch. Sister of Henry (Ruth) Piper and John (Penny) Piper. Also numerous nieces, nephews, and the very large extended family whom love Virginia dearly.
Donald C. "Pops" Vitous
Donald C. "Pops" Vitous, 98, of North Versailles Twp. passed away peacefully at home on Wednesday August 5, 2020.
Beloved husband of the late Anna (Fecho) Vitous. Loving father of John Vitous, Donald (Carol) Vitous, Jeannie (Carl) Krzywiec, James (Ayu) Vitous, Curtis (Anita) Vitous, and the late Timothy M. Vitous; 11 grandchildren, two great grandchildren and many beloved nieces, nephews, neighbors and friends.

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text:
Q: Resizing UIImageView clips sublayer I have a custom UIImageView which inserts a sublayer like this:
- (id)initWithFrame:(CGRect)frame
{
    self = [super initWithFrame:frame];
    if (self) {
        small_ = frame;
        float newX = self.center.x - ((self.frame.size.width*SELECTED_SCALE_VALUE)*0.5);
        float newY = self.center.y - ((self.frame.size.height*SELECTED_SCALE_VALUE)*0.5);
        big_ = CGRectMake(newX, newY, self.frame.size.width*SELECTED_SCALE_VALUE, self.frame.size.height*SELECTED_SCALE_VALUE);

        [self setupSubviews];
    }
    return self;
}

- (void)setupSubviews
{
    shadowImage_ = [[UIImage imageNamed:@"gallery_shadow_frame"] resizableImageWithCapInsets:UIEdgeInsetsMake(20, 20, 20, 20)];
    shadowImageView_ = [[UIImageView alloc] initWithFrame:[self shadowFrame]];
    shadowImageView_.image = shadowImage_;
    shadowImageView_.backgroundColor = [UIColor blueColor];
    [self.layer insertSublayer:[shadowImageView_ layer] atIndex:0];
}

When I later resize the frame of the UIImageView it clips the sublayer:
- (void)makeBig
{
    self.frame = big_;
    // shadowImageView_.frame = [self shadowFrame];
}

- (void)makeSmall
{
    self.frame = small_;
    // shadowImageView_.frame = [self shadowFrame];
}

To give a complete picture, there is also this:
- (CGRect)shadowFrame
{
    int x = -X_BORDER;
    int y = -Y_BORDER;
    int width = self.frame.size.width + X_BORDER * 2;
    int height = self.frame.size.height + Y_BORDER * 2;
    return CGRectMake(x, y, width, height);
}

How do I resize the UIImageView, but prevent it from clipping it's shadowImageView_ sublayer?

A: Have you tried
self.layer.masksToBounds = NO;

?


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This may be the year of the virtual currency. Virtual currencies have garnered much media, investor, and government attention. But, as venture capital funding continues to pour into the industry—more than $1 billion is expected to be invested this year alone—concerns remain over how the industry should deal with matters such as consumer protection and virtual currency's link to illegal activities. Unsurprisingly, state regulators have taken notice.
The first comprehensive regulatory framework in the United States will likely come from the New York State Department of Financial Services (NYDFS). The latest iteration of the Department's "BitLicense" applies to any entity engaged in a "virtual currency business activity," and would require that entity to register with the NYDFS. The revised BitLicense, however, narrowed the definition of "virtual currency" to exclude gift cards, digital units used solely in online games, and customer rewards programs. "Virtual currency business activity" was also limited, excluding entities that transact in nominal amounts of virtual currency or do so for non-financial purposes.
The revised BitLicense also relaxed, among other provisions, its licensing process and capital reserve requirements. Nevertheless, some industry stakeholders have expressed concern regarding the extent of the BitLicense's recordkeeping and customer identification provisions.
California also introduced virtual currency legislation. The proposed legislation contains a registration requirement similar to the BitLicense, but does not include anti-money laundering or know your customer provisions. However, it would require covered entities to maintain certain "high-quality, investment-grade permissible investments" equal to the sum of the virtual currency maintained on deposit for the licensee's customers.
In late May, the Digital Currency Jobs Creation Act was introduced in New Jersey. The bill would create a regulatory framework similar to the BitLicense, requiring entities that engage in a "digital currency custodial activity" to register with State. While the bill would impose certain cybersecurity, recordkeeping, and capital reserve requirements, it also offers incentives to regulated entities, including tax exemptions.
Commentators expect to see increasing legislative activity in this arena. New laws and regulations will likely have a profound effect on the industry, and should be monitored by interested parties. Given lawmakers' rapidly developing interest in virtual currencies, 2015 may prove to be the defining year for this nascent technology.


text:
\section{Introduction}

In this paper we study the geometry of the Kobasyashi distance $K_{\Omega}$ on domains $\Omega \subset \Cb^d$. Much is known about the behavior of the infinitesimal Kobayashi metric on certain types of domains (see for instance~\cite{JP2013} and the references therein), but very little is known about the global behavior of the Kobayashi distance function.

It is well known that the unit ball endowed with the Kobayashi metric is isometric to complex hyperbolic space and in particular is an example of a negatively curved Riemannian manifold. One would then suspect that when $\Omega \subset \Cb^d$ is a domain close to the unit ball then $(\Omega, K_\Omega)$ should be negatively curved in some sense. Unfortunately, for general domains the Kobayashi metric is no longer Riemannian and thus will no longer have curvature in a local sense. Instead one can ask if the Kobayashi metric satisfies a coarse notion of negative curvature from geometric group theory called Gromov hyperbolicity.

Gromov hyperbolic metric spaces have been intensively studied and have a number of remarkable properties. For instance:
\begin{enumerate}
\item Iterations of contractions on Gromov hyperbolic metric spaces are very well understood and in particular an analogue of the Wolff--Denjoy theorem always holds~\cite{K2001},
\item Given a quasi-isometry $f: X \rightarrow Y$ between two proper geodesic Gromov hyperbolic metric spaces there exists a continuous extension to natural compactifications of $X$ and $Y$ (see for instance~\cite[Chapter III.H, Theorem 3.9]{BH1999}),
\item (Geodesic shadowing) Every quasi-geodesic is within a bounded distance of an actual geodesic (see for instance~\cite[Chapter III.H, Theorem 1.7]{BH1999}).
\end{enumerate}
In particular, understanding the domains for which the Kobayashi metric is Gromov hyperbolic could lead to new insights about the iteration theory of holomorphic maps and the boundary extension properties of proper holomorphic maps. We should also mention that for the Kobayashi metric it is often easy to construct quasi-geodesics (see for instance Lemma~\ref{lem:quasi_geod} below) but difficult to find actual geodesics. Thus the geodesic shadowing property mentioned above can be a useful tool in understanding the Kobayashi distance function.

Balogh and Bonk~\cite{BB2000} proved that the Kobayashi metric is Gromov hyperbolic when the domain is strongly pseudo-convex. Gaussier and Seshadri~\cite{GS2013} and Nikolov, Thomas, and Trybula~\cite{NTT2014} proved, under certain boundary regularity conditions, that a non-trivial complex affine disk in the boundary of a convex set is an obstruction to the Gromov hyperbolicity of the Kobayashi metric. Extending this work, we recently characterized the bounded convex domains with smooth boundary for which the Kobayashi metric is Gromov hyperbolic:

\begin{theorem}\label{thm:finite_type}\cite{Z2014}
Suppose $\Omega$ is a bounded convex open set with $C^\infty$ boundary. Then $(\Omega, K_{\Omega})$ is Gromov hyperbolic if and only if $\partial \Omega$ has finite type in the sense of D'Angelo.
\end{theorem}

In this paper we explore what happens when convexity is relaxed to $\Cb$-convexity. Recall that an open set $\Omega \subset \Cb^d$ is called \emph{$\Cb$-convex} if its intersection with any complex line is either empty or contractible. See~\cite{APS2004} for the basic properties of these sets.

Estimates on the infinitesimal Kobayashi metric were established for $\Cb$-convex sets in~\cite{NPZ2011}. In particular, the Bergman, Carath{\'e}odory, and Kobayashi metrics are all bi-Lipschitz~\cite[Proposition 1, Theorem 12]{NPZ2011} for $\Cb$-convex sets which do not contain any complex affine lines. Since Gromov hyperbolicity is an quasi-isometry invariant, the results of this paper could be stated for the Bergman or Carath{\'e}odory metrics instead of the Kobayashi metric.

It has been suggested by several authors~\cite{BB2000, B2014, GS2013} that ``flatness'' in the boundary is an obstruction to the Kobayashi metric being Gromov hyperbolic. When the domain in question is convex this is indeed the case:

\begin{theorem}\label{thm:no_flats}\cite{GS2013, NTT2014, Z2014}
Suppose $\Omega \subset \Cb^d$ is a bounded convex open set and $(\Omega, K_{\Omega})$ is Gromov hyperbolic. If $\Delta \subset \Cb$ is the unit disk, then every holomorphic map $\varphi: \Delta \rightarrow \partial \Omega$ is constant.
\end{theorem}

\begin{remark} \ \begin{enumerate}
\item Theorem~\ref{thm:no_flats} was proven when $\Omega$ is bounded and $\partial \Omega$ is $C^\infty$ in~\cite{GS2013}, when $\partial \Omega$ is $C^{1,1}$ and $d=2$ in~\cite{NTT2014}, and in full generality in~\cite{Z2014}.
\item By~\cite{FS1998}, if $\Omega$ is a convex set then $\partial \Omega$ contains a non-trivial complex affine disk if and only if $\partial \Omega$ contains a non-trivial holomorphic disk.
\item The above theorem also holds for convex open sets which do no not contain any complex affine lines, see~\cite{Z2014}.
\end{enumerate}
 \end{remark}

The first part of this paper is devoted to proving an extension of Theorem~\ref{thm:no_flats} for bounded $\Cb$-convex sets whose boundary is $C^1$.

\begin{theorem}\label{thm:no_flats_C-convexity}
Suppose $\Omega \subset \Cb^d$ is a bounded $\Cb$-convex open set, $\partial \Omega$ is a $C^1$ hypersurface, and $(\Omega, K_{\Omega})$ is Gromov hyperbolic. If $\Delta \subset \Cb$ is the unit disk, then every holomorphic map $\varphi: \Delta \rightarrow \partial \Omega$ is constant.
\end{theorem}

\begin{remark} \ \begin{enumerate}
\item In Subsection~\ref{subsec:bd_domain} we will construct examples of bounded $\Cb$-convex sets $\Omega$ where $(\Omega, K_{\Omega})$ is Gromov hyperbolic but $\partial \Omega$ contains a complex affine ball of codimension one, thus showing that some boundary regularity is required.
\item By~\cite[Proposition 7]{NPZ2011}, if $\Omega$ is a bounded $\Cb$-convex open set with $C^1$ boundary then $\partial \Omega$ contains a non-trivial complex affine disk if and only if $\partial \Omega$ contains a non-trivial holomorphic disk.
\end{enumerate}
\end{remark}

In the second part of this paper we will demonstrate some new examples of unbounded convex domains for which the Kobayashi metric is Gromov hyperbolic. By taking projective images of these sets we will obtain bounded $\Cb$-convex sets. Our main motivation for these constructions is Proposition~\ref{prop:c_convex_disks} below, which shows that for any $d > 2$ there is a bounded $\Cb$-convex open set $\Omega \subset \Cb^d$ such that $(\Omega, K_{\Omega})$ is Gromov hyperbolic and $\partial \Omega$ contains a complex affine ball of dimension $d-1$.

Suppose $F:\Cb^d \rightarrow \Rb_{\geq 0}$ is a continuous function which is $C^\infty$ on $\Cb^d \setminus \{0\}$, non-negative, convex, and $F(0)=0$. Define
\begin{align*}
\Omega_F := \{ (z_0, \dots, z_d) \in \Cb^{d+1} : \Imaginary(z_0) > F(z_1, \dots, z_d) \}
\end{align*}
and
\begin{align*}
r_F(z_0,\dots, z_d) = F(z_1, \dots, z_d)-\Imaginary(z_0).
\end{align*}
We say that $\Omega_F$ has \emph{finite type away from 0} if for each non-trivial affine line $\ell:\Cb \rightarrow \Cb^{d+1}$ with $\ell(0) \in \partial \Omega_F \setminus \{0\}$ there exists multi-indices $\alpha,\beta$ such that
\begin{align*}
\frac{\partial^{\abs{\alpha}+\abs{\beta}}}{\partial z^{\alpha} \partial \overline{z}^{\beta}} (r_F \circ \ell)(0) \neq 0.
\end{align*}
If $\delta_1, \dots, \delta_d > 0$, we say that $F$ is \emph{$(\delta_1,\dots, \delta_d)$-homogeneous} if
\begin{align*}
\frac{1}{t} F( t^{\delta_1} z_1, \dots, t^{\delta_d} z_d) = F(z_1,\dots, z_d)
\end{align*}
for every $t >0$ and $(z_1,\dots, z_d) \in \Cb^d$. This implies that  $\Omega_F$ is invariant under the group
\begin{align*}
G=\left\{\begin{pmatrix}
t & & & \\
& t^{\delta_1} & & \\
& & \ddots & \\
& & & t^{\delta_d}
\end{pmatrix} : t > 0\right\}.
\end{align*}

Barth~\cite{B1980} proved that the Kobayashi metric is complete on a convex set $\Omega$ if and only if $\Omega$ does not contain any complex affine lines. Motivated by this fact and language from real projective geometry (see for instance~\cite{B2008}), we say a convex set $\Omega \subset \Cb^d$ is \emph{$\Cb$-proper} if $\Omega$ does not contain any complex affine lines.

Finally with all this language we will prove the following:

\begin{theorem}\label{thm:main}
Suppose $F:\Cb^d \rightarrow \Rb_{\geq 0}$ is a continuous function which is $C^\infty$ on $\Cb^d \setminus \{0\}$, non-negative, convex, and $F(0)=0$. If $\Omega_F$ is $\Cb$-proper, has finite type away from 0, and $F$ is $(\delta_1,\dots, \delta_d)$-homogeneous, then $(\Omega_F, K_{\Omega_F})$ is Gromov hyperbolic.
\end{theorem}

We should emphasize that for a general domain it is difficult to determine what geodesics in the Kobayashi metric look like. Thus establishing that the Kobayashi metric is Gromov hyperbolic is a non-trivial task and prior to the work in~\cite{Z2014} was (to the best of our knowledge) only established for bounded strongly pseudo-convex sets~\cite{BB2000} and for certain complex ellipses~\cite{GS2013}. The proof of Theorem~\ref{thm:main} relies heavily on the techniques used in~\cite{Z2014}.

\subsection{Homogeneous polynomial domains} Theorem~\ref{thm:main} applies to convex homogeneous polynomial domains. We say a convex polynomial $P:\Cb^d \rightarrow \Rb_{\geq 0}$ is \emph{non-degenerate} if $P^{-1}(0)$ does not contain any complex affine lines. Theorem~\ref{thm:main} then implies the following:

\begin{theorem}\label{thm:poly}
Suppose $P: \Cb^d \rightarrow \Rb$ is a non-negative, non-degenerate, convex, homogeneous polynomial with $P(0)=0$. If
\begin{align*}
\Omega = \{ (z_0, z_1, \dots, z_d) \in \Cb^{d+1} : \Imaginary(z_0) > P(z_1, \dots, z_d) \},
\end{align*}
then $(\Omega, K_{\Omega})$ is Gromov hyperbolic.
\end{theorem}

\subsection{Cones}

Theorem~\ref{thm:main} also applies to many convex cones. Suppose $F:\Cb^d \rightarrow \Rb_{\geq 0}$ is a continuous function which is $C^\infty$ on $\Cb^d \setminus \{0\}$, non-negative, convex, and $F(0)=0$. If
\begin{align*}
\frac{1}{t}F(t z_1, \dots, tz_d) = F(z_1, \dots, z_d)
\end{align*}
for all $z_1, \dots, z_d \in \Cb$ and $t > 0$, then $\Omega_F$ is a convex cone. And if $\Omega_F$ is a $\Cb$-proper and has finite type away from 0, then $\Omega_F$ satisfies the hypothesis of Theorem~\ref{thm:main} .

To give a concrete example of this construction let $\norm{z}_p$ be the $L_p$-norm on $\Cb^d$ then:

\begin{theorem}\label{thm:cones} If $p > 1$ and
\begin{align*}
\Cc_p = \{ (z_0, z) \in \Cb^{d+1} : \Imaginary(z_0) > \norm{z}_p  \},
\end{align*}
then $(\Cc_p, K_{\Cc_p})$ is Gromov hyperbolic.
\end{theorem}

\subsection{Bounded domains}\label{subsec:bd_domain}

We can also use Theorem~\ref{thm:main} to constructed examples of bounded sets where the Kobayashi metric is Gromov hyperbolic. This can be accomplished by taking the image of one of the domains above under an appropriate projective transformation. Unfortunately convexity is not preserved under such maps and instead the resulting domains are only $\Cb$-convex.

For instance, consider the cone $\Cc_2$ from Theorem~\ref{thm:cones} and the transformation $f : \Cb^{d+1} \setminus \{ z_0=-i\} \rightarrow \Cb^{d+1}$ given by
\begin{align*}
f(z_0,\dots,  z_d) = \left( \frac{1}{z_0+i},\frac{z_1}{z_0+i},\dots, \frac{z_d}{z_0+i} \right).
\end{align*}
Notice that $f$ is a restriction of a projective automorphism $\Pb(\Cb^{d+2}) \rightarrow \Pb(\Cb^{d+2})$. In particular $\Omega:=f(\Cc_2)$ is bi-holomorphic to $\Cc_2$ and hence $(\Omega, K_{\Omega})$ is Gromov hyperbolic. Since $\Cc_2$ is convex and $f$ is a projective automorphism the intersection of $\Omega$ with any complex line is either empty or simply connected. Thus $\Omega$ is a  $\Cb$-convex set. Moreover $\Omega$ is bounded and the boundary of $\Omega$ contains $\{0\} \times \{ (z_1, \dots, z_d) : \sum \abs{z_i}^2 < 1\}$. Summarizing the above example:

\begin{proposition}\label{prop:c_convex_disks}
For any $d \geq 2$, there is a bounded $\Cb$-convex open set $\Omega \subset \Cb^d$ such that $(\Omega, K_{\Omega})$ is Gromov hyperbolic and $\partial \Omega$ contains a complex affine ball of dimension $d-1$.
\end{proposition}

\subsection{Corners and the Hilbert metric}
Every proper open convex set $\Omega \subset \Rb^d$ has a projectively invariant metric $H_{\Omega}$ called the \emph{Hilbert metric}. This metric is usually defined using cross ratios, but it has an equivalent formulation which makes it a real projective analogue of the Kobayashi metric (see for instance~\cite{K1977} or~\cite{L1986} or~\cite[Section 3.4]{G2015}). Thus results about the Hilbert metric can serve as guide to understanding the Kobayashi metric.

The convex domains for which the Hilbert metric is Gromov hyperbolic are very well understood. Karlsson and Noskov showed:

\begin{theorem}\cite{KN2002}\label{thm:hilbert}
If $\Omega \subset \Rb^d$ is a convex set and $(\Omega, H_{\Omega})$ is Gromov hyperbolic, then $\partial \Omega$ is a $C^1$ hypersurface and $\Omega$ is strictly convex (that is $\partial \Omega$ does not contain any line segments).
\end{theorem}

\begin{remark}\label{rmk:hilbert} Notice that $\partial\Omega$ being $C^1$ is a equivalent to there being a unique supporting real hyperplane through each boundary point and $\Omega$ being strictly convex is equivalent to each supporting real hyperplane intersecting $\partial \Omega$ at exactly one point.
\end{remark}

Improving on Theorem~\ref{thm:hilbert}, Benoist~\cite{B2003} characterized the convex domains for which the Hilbert metric is Gromov hyperbolic in terms of the first derivatives of local defining functions for $\partial \Omega$.

It is natural to ask if some analogue of Theorem~\ref{thm:hilbert} holds for the Kobayashi metric. Since every bounded convex set in $\Cb$ has Gromov hyperbolic Kobayashi metric, in general $\partial \Omega$ need not be $C^1$ and $\Omega$ need not be strictly convex.  However the conclusion of Theorem~\ref{thm:no_flats} can be seen as a complex analytic version of strict convexity.

Based on Theorem~\ref{thm:hilbert}, Remark~\ref{rmk:hilbert}, and Theorem~\ref{thm:no_flats} it is natural to ask if the number of complex supporting hyperplanes through a point in the boundary is restricted by the Gromov hyperbolicity of the Kobayashi metric. However if $f: \Cb^d \rightarrow \Cb$ is a linear map and $\abs{f(z)} \leq \abs{z}$ for all $z \in \Cb^d$ then the complex hyperplane
\begin{align*}
H_{f} = \{ (f(z), z) : z \in \Cb^{d}\}
\end{align*}
is tangent to the cone $\Cc_2$ (in Theorem~\ref{thm:cones}) at $0$. In particular we see that the set of supporting complex hyperplanes of $\Cc_2$ through $0$ contains a complex ball of dimension $d$.

\subsection*{Acknowledgments}

I would like to thank the referee for a number of comments and corrections which greatly improved the present work.

\section{Preliminaries}\label{sec:prelim}

\subsection{Basic notation} We now fix some very basic notations.

\begin{itemize}
\item Let $\Delta: = \{ z \in \Cb : \abs{z} < 1\}$.
\item For $z \in \Cb^d$ let $\norm{z}$ denote the standard Euclidean norm of $z$.
\item For $z_0 \in \Cb^d$ and $R>0$ let $B_R(z_0) := \{ z \in \Cb^d : \norm{z-z_0} < R\}$.
\item Given a open set $\Omega \subset \Cb^d$ and $p \in \Omega$ let
\begin{align*}
\delta_{\Omega}(p):= \inf \left\{ \norm{q-p} : q \in  \partial \Omega \right\}.
\end{align*}
\item Given a open set $\Omega \subset \Cb^d$, $p \in \Omega$, and $v \in \Cb^d$ let
\begin{align*}
\delta_{\Omega}(p;v):= \inf \left\{ \norm{q-p} : q \in (p+\Cb \cdot v)  \cap \partial \Omega \right\}.
\end{align*}
\item Given two open sets $\Omega_1 \subset \Cb^{d_1}$ and $\Omega_2 \subset \Cb^{d_2}$ let $\Hol(\Omega_1, \Omega_2)$ be the space of holomorphic maps from $\Omega_1$ to $\Omega_2$.
\item Given two open sets $\Oc_1 \subset \Rb^{d_1}$, $\Oc_2 \subset \Rb^{d_2}$, a $C^1$ map $F: \Oc_1 \rightarrow \Oc_2$, and a point $x \in \Oc_1$ define the derivative $d(F)_x : \Rb^{d_1} \rightarrow \Rb^{d_2}$ of $F$ at $x$ by
\begin{align*}
d(F)_x(v) := \left.\frac{d}{dt}\right|_{t=0} F(x+tv).
\end{align*}
\end{itemize}

\subsection{The Kobayashi metric and distance} Given a domain $\Omega \subset \Cb^d$ the \emph{(infinitesimal) Kobayashi metric} is the pseudo-Finsler metric
\begin{align*}
k_{\Omega}(x;v) = \inf \left\{ \abs{\xi} : f \in \Hol(\Delta, \Omega), \ f(0) = x, \ d(f)_0(\xi) = v \right\}.
\end{align*}
By a result of Royden~\cite[Proposition 3]{R1971} the Kobayashi metric is an upper semicontinuous function on $\Omega \times \Cb^d$. In particular if $\sigma:[a,b] \rightarrow \Omega$ is an absolutely continuous curve (as a map $[a,b] \rightarrow \Cb^d$), then the function
\begin{align*}
t \in [a,b] \rightarrow k_\Omega(\sigma(t); \sigma^\prime(t))
\end{align*}
is integrable and we can define the \emph{length of $\sigma$} to  be
\begin{align*}
\ell_\Omega(\sigma)= \int_a^b k_\Omega(\sigma(t); \sigma^\prime(t)) dt.
\end{align*}
One can then define the \emph{Kobayashi pseudo-distance} to be
\begin{multline*}
 K_\Omega(x,y) = \inf \left\{\ell_\Omega(\sigma) : \sigma\colon[a,b]
 \rightarrow \Omega \text{ is absolutely continuous}, \right. \\
 \left. \text{ with } \sigma(a)=x, \text{ and } \sigma(b)=y\right\}.
\end{multline*}
This definition is equivalent to the standard definition of $K_\Omega$ via analytic chains, see~\cite[Theorem 3.1]{V1989}.

A nice introduction to the Kobayashi metric and its basic properties can be found in~\cite{K2005}.

\subsection{Gromov hyperbolic metric spaces}\label{sec:prelim_gromov}

Suppose $(X,d)$ is a metric space. If $I \subset \Rb$ is an interval, a curve $\sigma: I \rightarrow X$ is a \emph{geodesic} if $d(\sigma(t_1),\sigma(t_2)) = \abs{t_1-t_2}$ for all $t_1, t_2 \in I$.  A \emph{geodesic triangle} in a metric space is a choice of three points in $X$ and geodesic segments  connecting these points. A geodesic triangle is said to be \emph{$\delta$-thin} if any point on any of the sides of the triangle is within distance $\delta$ of the other two sides.

\begin{definition}
A proper geodesic metric space $(X,d)$ is called \emph{$\delta$-hyperbolic} if every geodesic triangle is $\delta$-thin. If $(X,d)$ is $\delta$-hyperbolic for some $\delta\geq0$ then $(X,d)$ is called \emph{Gromov hyperbolic}.
\end{definition}

The book by Bridson and Haefliger~\cite{BH1999} is one of the standard references for Gromov hyperbolic metric spaces.

\section{A lower bound for the Kobayashi distance}

In this section we use an estimate for the  infinitesimal Kobayashi metric established by Nikolov, Pflug, and Zwonek ~\cite{NPZ2011} to obtain an estimate on the Kobayashi distance. Using this estimate on the distance we will demonstrate a basic property of the asymptotic geometry of the Kobayashi distance on $\Cb$-convex sets.

By considering affine maps of the unit disk into a domain, one obtains the following upper bound on the Kobayashi metric:

\begin{observation}
 Suppose $\Omega \subset \Cb^d$ is an open set. Then
\begin{align*}
k_{\Omega}(p;v) \leq \frac{\norm{v}}{\delta_{\Omega}(p;v)}
\end{align*}
for $p \in \Omega$ and $v \in \Cb^d$ non-zero.
\end{observation}

For $\Cb$-convex sets Nikolov, Pflug, and Zwonek obatined a lower bound on the Kobayashi metric:

\begin{proposition}\label{prop:convex_lower_bd_1}\cite[Proposition 1]{NPZ2011}
Suppose $\Omega \subset \Cb^d$ is an open $\Cb$-convex set. Then
\begin{align*}
\frac{\norm{v}}{4\delta_{\Omega}(p;v)} \leq k_{\Omega}(p;v)
\end{align*}
for $p \in \Omega$ and $v \in \Cb^d$ non-zero.
 \end{proposition}

 Using Proposition~\ref{prop:convex_lower_bd_1} we can obtain a lower bound for the Kobayashi distance.

\begin{lemma}\label{lem:convex_lower_bd_2}
Suppose $\Omega \subset \Cb^d$ is an open $\Cb$-convex set and $p,q \in \Omega$ are distinct. Let $L$ be the complex line containing $p$ and $q$. If $\xi \in L \setminus L \cap \Omega$, then
\begin{align*}
 \frac{1}{4}\abs{\log \left( \frac{\norm{q-\xi}}{\norm{p-\xi}} \right)} \leq K_{\Omega}(p,q).
 \end{align*}
 \end{lemma}

 \begin{remark} For our purposes the above estimate suffices, but the more precise estimate
 \begin{align*}
  \frac{1}{4} \log \left(1+  \frac{\norm{p-q}}{\min\{ \delta_\Omega(p; q-p), \delta_\Omega(q; p-q)\} } \right) \leq K_{\Omega}(p,q)
  \end{align*}
  follows from the proof of Proposition 2 part (ii) in~\cite{NT2015}.
 \end{remark}

\begin{proof}
Since $p,q, \xi$ are all contained in a single affine line the quantity
\begin{align*}
\abs{\log \left( \frac{\norm{q-\xi}}{\norm{p-\xi}} \right)}
 \end{align*}
 is invariant under affine transformations. In particular we can assume that $\xi=0$, $p=(p_1,0,\dots,0)$, and $q = (q_1,0,\dots, 0)$. Since $\Omega$ is $\Cb$-convex there exists an complex hyperplane $H$ such that $0 \in H$ but $H \cap \Omega = \emptyset$ (see~\cite[Theorem 2.3.9]{APS2004}). Using another affine transformation we may assume in addition that
 \begin{align*}
 H=\{ (0,z_1, \dots, z_{d-1}) \in \Cb^d : z_1, \dots, z_{d-1} \in \Cb\}.
 \end{align*}
 Now consider the projection $P:\Cb^d \rightarrow \Cb$ onto the first component. Then
 \begin{align*}
 K_{\Omega}(p,q) \geq K_{P(\Omega)}(p_1,q_1).
 \end{align*}
 By~\cite[Theorem 2.3.6]{APS2004} the linear image of a $\Cb$-convex set is $\Cb$-convex. Hence $P(\Omega) \subset \Cb$ is a $\Cb$-convex open set. Since $P^{-1}(0)=H$ we see that $P(\Omega)$ does not contain zero. Now suppose that $\sigma: [0,1] \rightarrow P(\Omega)$ is a absolutely continuous curve with $\sigma(0) = p_1$ and $\sigma(1)=q_1$. Then since $0 \in \Cb \setminus P(\Omega)$, Proposition~\ref{prop:convex_lower_bd_1} implies that
 \begin{align*}
 \int_0^1 k_{P(\Omega)}(\sigma(t); \sigma^\prime(t)) dt
 &\geq \frac{1}{4} \int_0^1 \frac{ \abs{\sigma^\prime(t)}}{\abs{\sigma(t)}} dt \geq \frac{1}{4} \int_0^1 \frac{ \abs{\frac{d}{dt}\abs{\sigma(t)}}}{\abs{\sigma(t)}} dt \\
 & \geq \frac{1}{4}\abs{ \int_0^1 \frac{ \frac{d}{dt}\abs{\sigma(t)}}{\abs{\sigma(t)}} dt } \geq\frac{1}{4}\abs{ \log \left( \frac{\abs{q_1}}{\abs{p_1}} \right)} \\
 & =  \frac{1}{4}\abs{\log \left( \frac{\norm{q-\xi}}{\norm{p-\xi}} \right)}.
 \end{align*}
 Since $\sigma$ was an arbitrary absolutely continuous curve joining $p_1$ to $q_1$ the Lemma follows.
 \end{proof}

 Using Lemma~\ref{lem:convex_lower_bd_2} we can obtain some information about the asymptotic geometry of the Kobayashi distance.

 \begin{proposition}\label{prop:asym_geom}
Suppose $x, y \in \partial \Omega$ are distinct and $(p_n)_{n \in \Nb}, (q_m)_{m \in \Nb} \subset \Omega$ are sequences such that $p_n \rightarrow x$, $q_m \rightarrow y$, and
\begin{align*}
\liminf_{n,m \rightarrow \infty} K_{\Omega}(p_n, q_m) < \infty.
\end{align*}
If $L$ is the complex line containing $x$ and $y$, then the interior of $\overline{\Omega} \cap L$ in $L$ contains $x$ and $y$.
\end{proposition}

\begin{proof}
By passing to subsequences we may suppose that there exists $M < \infty$ such that
\begin{align*}
K_{\Omega}(p_n, q_n) < M
\end{align*}
for all $n \in \Nb$. For each $n$, let $L_n$ be the complex affine line containing $p_n$ and $q_n$. Let
\begin{align*}
\epsilon_n = \min\{\norm{\xi-p_n} : \xi \in  L_n \setminus \Omega \cap L_n\}
\end{align*}
and $\xi_n \in L_n \setminus \Omega \cap L_n$ be a point closest to $p_n$. Then by Lemma~\ref{lem:convex_lower_bd_2}
\begin{align*}
M &\geq \limsup_{n \rightarrow \infty} K_{\Omega}(p_n, q_n) \geq \limsup_{n \rightarrow \infty} \frac{1}{4} \log \frac{ \norm{q_n - \xi_n}}{\norm{p_n - \xi_n}} \\
& \geq   \limsup_{n \rightarrow \infty} \frac{1}{4} \log \frac{ \norm{q_n -p_n}- \epsilon_n}{\epsilon_n} \\
& \geq   \limsup_{n \rightarrow \infty} \frac{1}{4} \log \frac{ \norm{x -y}- \epsilon_n}{\epsilon_n}.
\end{align*}
Since $x \neq y$ there exists an $\epsilon>0$ such that $B_{\epsilon}(p_n) \cap L_n \subset L_n \cap \Omega$ for all $n$ sufficiently large. Which implies that $x$ is in the interior of $\overline{\Omega} \cap L$ in $L$. The same argument applies to $y$.
\end{proof}

\section{Proof of Theorem~\ref{thm:no_flats_C-convexity}}

We begin with a sketch of the argument: \newline

\textbf{Idea of Proof:} \emph{We assume, for a contradiction, that $(\Omega, K_{\Omega})$ is Gromov hyperbolic and there exists a non-constant holomorphic map $\varphi: \Delta \rightarrow \partial \Omega$. Since $\partial \Omega$ is $C^1$ the existence of a non-constant holomorphic disk in the boundary implies the existence of a non-trivial affine disk~\cite[Proposition 7]{NPZ2011}. We will show that every inward pointing normal line of $\partial \Omega$ can be parameterized to be a quasi-geodesic and if two such quasi-geodesics terminate in the same non-trivial open affine disk in the boundary then they stay within a uniform bounded distance of each other. We then take limits to show that this also holds for two such quasi-geodesics which terminate in the same non-trivial closed affine disk in the boundary. But if the closed disk is maximal this will contradict Proposition~\ref{prop:asym_geom}.} \newline

If $\Omega \subset \Cb^d$ is an open set, $\partial \Omega$ is a $C^1$ hypersurface, and $x \in \partial \Omega$ let $T_x \partial\Omega \subset \Cb^d$ be the real hyperplane tangent to $\partial \Omega$ at $x$ and let $n_x \in \Cb^d$ be the inward pointing unit normal vector at $x$.

We will repeatedly use the following observation in the proof of Theorem~\ref{thm:no_flats_C-convexity}:

\begin{observation}\label{obs:C1_domains}
Assume $\Omega \subset \Cb^d$ is a bounded open set and $\partial \Omega$ is a $C^1$ hypersurface. Then:
\begin{enumerate}
\item There exists $C,\epsilon_1 > 0$ so that
\begin{align*}
\delta_\Omega(x+tn_x) \geq Ct
\end{align*}
for any $x \in \partial \Omega$ and $t \in (0,\epsilon_1)$.
\item For any $R > 0$ there exists $\epsilon_2 > 0$ so that
\begin{align*}
\delta_\Omega(x+tn_x; v) \geq Rt
\end{align*}
for any $x \in \partial \Omega$, $v \in T_x\partial \Omega$, and $t \in (0,\epsilon_2)$.
\end{enumerate}
\end{observation}

\subsection{Quasi-geodesics in Gromov hyperbolic metric spaces}

\begin{definition}
Suppose $(X,d)$ is a metric space, $A \geq 1$, and $B \geq 0$. If $I \subset \Rb$ is an interval, then a map $\sigma: I \rightarrow X$ is a \emph{$(A,B)$-quasi-geodesic} if
\begin{align*}
\frac{1}{A} \abs{t-s} - B \leq d(\sigma(s), \sigma(t)) \leq A \abs{t-s} + B
\end{align*}
for all $s,t \in I$.
\end{definition}

We will need a basic property of quasi-geodesics in Gromov hyperbolic metric spaces.

\begin{proposition}\label{prop:gromov_basic}
Suppose $(X,d)$ is a proper geodesic Gromov hyperbolic metric space. For any $A \geq 1$ and $B \geq 0$ there exists $M \geq 0$ such that if $\sigma_1,\sigma_2: \Rb_{\geq 0} \rightarrow X$ are $(A,B)$-quasi-geodesics and
\begin{align*}
\liminf_{t \rightarrow \infty} d(\sigma_1(t), \sigma_2) < \infty,
\end{align*}
then
\begin{align*}
\sup_{t \geq 0} \Big( \max\left\{  d(\sigma_1(t), \sigma_2), \ d(\sigma_2(t), \sigma_1)\right\}\Big) \leq M+2d(\sigma_1(0), \sigma_2(0)).
\end{align*}

\end{proposition}

Before starting the proof of the proposition we will make one observation, but first some notation: A \emph{geodesic rectangle} in a metric space $(X,d)$ is a choice of four geodesic segments
\begin{align*}
 \sigma_i : [a_i,b_i] \rightarrow X \quad i=1,2,3,4
\end{align*}
so that $\sigma_1(b_1) = \sigma_2(a_2)$, $\sigma_2(b_2) = \sigma_3(a_3)$, $\sigma_3(b_3) = \sigma_4(a_4)$, and $\sigma_4(b_4)=\sigma_1(a_1)$. A geodesic rectangle is said to be \emph{$\delta$-thin} if any point on any of the sides of the rectangle is within distance $\delta$ of the other three sides. By connecting a pair of opposite vertices in a geodesic rectangle by a geodesic and considering the resulting two geodesic triangles the following observation is immediate:

\begin{observation}
 Suppose $(X,d)$ is a proper geodesic $\delta$-hyperbolic metric space. Then every geodesic rectangle in $(X,d)$ is $(2\delta)$-thin.
\end{observation}

\begin{proof}[Proof of Proposition~\ref{prop:gromov_basic}]
Assume that $(X,d)$ is $\delta$-hyperbolic for some $\delta \geq 0$.

By~\cite[Chapter III.H, Theorem 1.7]{BH1999} for any $A \geq 1$ and $B \geq 0$ there exists $M_1 >0$ such that for any $(A,B)$-quasi-geodesic $\sigma: \Rb_{\geq 0} \rightarrow X$ there exists a geodesic $\overline{\sigma}:\Rb_{\geq 0} \rightarrow X$ such that $\sigma(0) = \overline{\sigma}(0)$ and
\begin{align*}
\sup_{t \geq 0} \Big(\max\{ d(\sigma(t), \overline{\sigma}), \ d(\overline{\sigma}(t), \sigma) \}\Big) \leq M_1.
\end{align*}
Thus the proposition reduces to the following claim: there exists $M \geq 0$ such that if $\sigma_1, \sigma_2 : \Rb_{\geq 0} \rightarrow X$ are two geodesic rays and
\begin{align*}
\liminf_{t \rightarrow \infty} K_{\Omega}(\sigma_1(t), \sigma_2) < \infty
\end{align*}
then
\begin{align*}
\sup_{t >0} d(\sigma_1(t), \sigma_2(t)) \leq M + 2d(\sigma_1(0), \sigma_2(0)).
\end{align*}
Pick a sequence $T_n \rightarrow \infty$ such that
\begin{align*}
\sup_{n \in \Nb} d(\sigma_1(T_n), \sigma_2)=C < \infty
\end{align*}
for some $C \geq 0$. Let $\gamma_0$ be a geodesic segment joining $\sigma_1(0)$ to $\sigma_2(0)$. Fix $n>0$ sufficiently large and let $\gamma_n$ be a geodesic joining $\sigma_1(T_n)$ to a closest point $\sigma_2(T_n^\prime)$ on $\sigma_2$.

Now the geodesic segments $\gamma_0$, $\sigma_1|_{[0,T_1]}$, $\gamma_n$, and $\sigma_2|_{[0,T_1^\prime]}$ form a geodesic rectangle which is $(2\delta)$-thin. But for
\begin{align*}
t \in \left[d(\sigma_1(0), \sigma_2(0)) + 2 \delta, T_n - (C + 2 \delta)\right]
\end{align*}
the point $\sigma_1(t)$ is not $(2\delta)$-close to either $\gamma_0$ or $\gamma_n$, hence there exists $t^\prime$ such that $d(\sigma_1(t), \sigma_2(t^\prime)) \leq 2 \delta$. Now
\begin{align*}
2 \delta \geq d(\sigma_1(t), \sigma_2(t^\prime)) \geq \abs{t^\prime - t} - d(\sigma_1(0), \sigma_2(0)).
\end{align*}
Hence $\abs{t^\prime - t} \leq 2 \delta + d(\sigma_1(0), \sigma_2(0))$ and so
\begin{align*}
d(\sigma_1(t), \sigma_2(t)) \leq 4\delta + d(\sigma_1(0), \sigma_2(0))
\end{align*}
for $t \in \left[d(\sigma_1(0), \sigma_2(0)) + 2 \delta, T_n - (C + 2 \delta)\right]$. Hence
\begin{align*}
d(\sigma_1(t), \sigma_2(t)) \leq 4\delta + 2d(\sigma_1(0), \sigma_2(0))
\end{align*}
for $t \leq T_n - (C + 2 \delta)$. Sending $n \rightarrow \infty$ proves the claim and thus the proposition.
\end{proof}

\subsection{Quasi-geodesics in $\Cb$-convex domains}

For $\Cb$-convex domains with $C^1$ boundary normal lines can be parametrized as quasi-geodesics:

\begin{lemma}\label{lem:quasi_geod}
Suppose $\Omega \subset \Cb^d$ is a bounded $\Cb$-convex open set and $\partial \Omega$ is a $C^1$ hypersurface. Then there exists $A \geq 1$ and $\epsilon >0$ so that for any $x \in \partial \Omega$ the curve
\begin{align*}
\sigma_{x}&: \Rb_{\geq 0} \rightarrow \Omega \\
\sigma_{x}&(t)=x+e^{-t}\epsilon n_x
\end{align*}
is a $(A,0)$-quasi-geodesic in $(\Omega, K_{\Omega})$.
\end{lemma}

\begin{proof}
Using Observation~\ref{obs:C1_domains} there exists $C, \epsilon >0$ so that
\begin{align*}
\delta_\Omega(x+tn_x) \geq Ct
\end{align*}
for any $x \in \partial \Omega$ and $t \in (0,\epsilon)$.

Now fix $x \in \partial \Omega$. By Lemma~\ref{lem:convex_lower_bd_2}
\begin{align*}
K_{\Omega}(\sigma_{x}(s),\sigma_{x}(t)) \geq  \frac{1}{4}\abs{ \log \frac{\norm{\sigma_{x}(t)-x}}{\norm{\sigma_{x}(s)-x}}} =  \frac{1}{4} \abs{t-s}.
\end{align*}
And if $s \leq t$ then
\begin{align*}
K_{\Omega}(\sigma_{x}(s),\sigma_{x}(t))
&\leq \int_{s}^t k_{\Omega}(\sigma_{x}(r); \sigma_{x}^\prime(r)) dr
\leq \int_s^t \frac{ \norm{\sigma_{x}^\prime(r)} dr}{\delta_{\Omega}(\sigma_{x}(r))} \\
& \leq  \int_s^t \frac{ e^{-r}\epsilon }{Ce^{-r}\epsilon} dr
 \leq \frac{1}{C}\abs{t-s}.
\end{align*}
So $\sigma_x$ is a $(A,0)$-quasi-geodesic where $A = \max \{ 4, 1/C\}$.
\end{proof}

\subsection{Normal lines in domains with $C^1$ boundary}

\begin{proposition}\label{prop:C1}
Assume $\Omega \subset \Cb^d$ is an open set and $\partial \Omega$ is a $C^1$ hypersurface. Suppose that $L$ is a complex affine line so that $L \cap \partial \Omega$ has non-empty interior in $L$. If $\Uc$ is a connected component of the interior of $L \cap \partial \Omega$ and $x,y \in \Uc$, then
\begin{align*}
\limsup_{t \searrow 0} K_\Omega(x+tn_x, y+tn_y) < \infty.
\end{align*}
\end{proposition}

We begin the proof of Proposition~\ref{prop:C1} with a lemma:

\begin{lemma} \label{lem:same_pt} Assume $\Omega \subset \Cb^d$ is an open set and $\partial \Omega$ is a $C^1$ hypersurface. If $x \in \partial \Omega$ and $v \in \Cb^d$ is a unit vector with $\angle(n_x, v) < \pi/2$, then
\begin{align*}
\limsup_{t \searrow 0} K_\Omega(x+tn_x, x+tv) < \infty.
\end{align*}
\end{lemma}

\begin{proof}
By hypothesis $v = \lambda n_x + v^\prime$ where $\lambda \in (0,1]$ and $\angle(v^\prime,n_x) =\pi/2$. Then
 \begin{align*}
K_\Omega(x+tn_x, x+tv) \leq K_\Omega(x+tn_x, x+t\lambda n_x) + K_\Omega(x+t\lambda n_x, x+tv).
\end{align*}
We will bound each term individually.

By Observation~\ref{obs:C1_domains} there exists $C, \epsilon_1 >0$ so that
\begin{align*}
\delta_\Omega(x+tn_x) \geq C t \text{ for } t \in (0,\epsilon_1).
\end{align*}
Then for $t \in (0,\epsilon_1)$, define the curve $\gamma_t: [\lambda,1] \rightarrow \Omega$ by
\begin{align*}
\gamma_t(s) = x+stn_x.
\end{align*}
Then
\begin{align*}
K_\Omega(x+tn_x, x+t\lambda n_x)
& \leq \int_{\lambda}^1 k_\Omega(\gamma_t(s); \gamma_t^\prime(s)) ds
\leq \int_{\lambda}^1 \frac{\norm{\gamma_t^\prime(s)}}{\delta_\Omega(\gamma_t(s))} ds  \\
& \leq \int_{\lambda}^1 \frac{1}{Cs} ds = -\frac{1}{C}\log(\lambda).
 \end{align*}

 Next let $H = \{ w \in \Cb^d : \angle(w,n_x) =\pi/2\}$. Then $x+H= T_x \partial \Omega$ and so by Observation~\ref{obs:C1_domains} there exists $\epsilon_2 > 0$ so that:
 \begin{align*}
 \delta_\Omega(x+tn_x; w) \geq \left(\frac{1+\norm{v^\prime}}{\lambda}\right)t
 \end{align*}
for all $w \in H$ and $t \in (0, \epsilon_2)$. Then for $t \in (0,\epsilon_2)$, define the curve $\sigma_t: [0,1] \rightarrow \Omega$ by
\begin{align*}
\sigma_t(s) = s(x + t\lambda n_x) + (1-s)(x+tv) = x+t\lambda n_x + t(1-s)v^\prime
\end{align*}
Now $\sigma_t^\prime(s) = -tv^\prime \in H$ and so
 \begin{align*}
 \delta_\Omega(\sigma_t(s); \sigma_t^\prime(s)) \geq t.
\end{align*}
Then
\begin{align*}
K_\Omega(x+t\lambda n_x, x+tv)
& \leq \int_{0}^1 k_\Omega(\sigma_t(s); \sigma_t^\prime(s)) ds
\leq \int_{0}^1 \frac{\norm{\sigma_t^\prime(s)}}{\delta_\Omega(\sigma_t(s); \sigma_t^\prime(s))} ds  \\
& \leq \int_{0}^1 \frac{t \norm{v^\prime}}{t} ds = \norm{v^\prime}.
 \end{align*}
Combining the two estimates above we obtain the lemma.
 \end{proof}

\begin{proof}[Proof of Proposition~\ref{prop:C1}] Since $\Uc$ is connected it is enough to show: for any $x \in \Uc$ there exists a neighborhood $\Vc$ of $x$ in $\Uc$ so that:
\begin{align*}
\limsup_{t \searrow 0} K_\Omega(x+tn_x, y+tn_y) < \infty \text{ for all } y \in \Vc.
\end{align*}
So fix $x \in \Uc$. By translating $\Omega$ by an affine isometry we can assume that: $x=0$, $n_x = (i,0, \dots, 0)$, $L=  \{ (0,z,0,\dots,0) : z \in \Cb\}$, and $T_x \partial \Omega= \Rb \times \Cb^{d-1}$. Then since $\partial \Omega$ is $C^1$ there exists $\epsilon >0$ and a $C^1$ function $F:  \Rb \times \Cb^{d-1} \rightarrow \Rb$ such that
\begin{align*}
\{ z \in \Cb^d : \norm{z}_\infty < \epsilon \} & \cap \Omega
=  \{ (u+iw, z_2, \dots, z_d) : w > F(u,z_2, \dots, z_d) \}.
\end{align*}
Now since $L=  \{ (0,z,0,\dots,0) : z \in \Cb\}$ we can decrease $\epsilon$ so that
\begin{align*}
F(0,z_2,0,\dots, 0) =0
\end{align*}
 for $\abs{z_2} < \epsilon$. This implies that
 \begin{align}
\label{eq:containment}
tn_x + \{ (0,z,0,\dots, 0) : \abs{z} < \epsilon\} \subset \Omega
\end{align}
when $t \in (0,\epsilon)$. Now let $\Vc = \{ (0,z,0,\dots, 0) : \abs{z} < \epsilon/2\}$. By decreasing $\epsilon$ we can assume that for any $y \in \Vc$ we have $\angle(n_y, n_x) < \pi/2$.

Now fix $y \in \Vc$. Then
\begin{align*}
\limsup_{t \searrow 0} K_\Omega(y+tn_x, y+tn_y) < \infty
\end{align*}
by Lemma~\ref{lem:same_pt} and so it is enough to show that
\begin{align*}
\limsup_{t \searrow 0} K_\Omega(x+tn_x, y+tn_x) < \infty.
\end{align*}
For $t \in (0,\epsilon)$ define the curve $\gamma_t : [0,1] \rightarrow \Omega$ by
\begin{align*}
\gamma_t(s) = s(x+tn_x) + (1-s)(y+tn_x) =(1-s)y + tn_x.
\end{align*}
Then the inclusion in~\ref{eq:containment} implies that
\begin{align*}
\delta_\Omega(\gamma_t(s));\gamma_t^\prime(s)) \geq \epsilon/2.
\end{align*}
Then for $t \in (0,\epsilon)$ we have
\begin{align*}
K_\Omega(x+t n_x, & y+tn_x) \leq  \int_0^1  k_\Omega(\gamma_t(s); \gamma_t^\prime(s)) ds \\
& \leq \int_0^1  \frac{\norm{\gamma_t^\prime(s)}}{\delta_\Omega(\gamma_t(s));\gamma_t^\prime(s)) } ds
\leq \frac{2 \norm{y}}{\epsilon}.
 \end{align*}
\end{proof}

\subsection{The proof of Theorem~\ref{thm:no_flats_C-convexity}}

Suppose for a contradiction that $\Omega$ is a bounded $\Cb$-convex domain, $\partial \Omega$ is a $C^1$ hypersurface, $(\Omega, K_{\Omega})$ is Gromov hyperbolic, and there exists a non-trivial holomorphic map $\varphi: \Delta \rightarrow \partial \Omega$.

By~\cite[Proposition 7]{NPZ2011} there exists an non-trivial affine map $\ell: \Delta \rightarrow \partial \Omega$. Let $L$ be the complex affine line containing $\ell(\Delta)$ and let $\Uc$ be a connected component of the interior of $L \cap \partial \Omega$ in $L$.

By Lemma~\ref{lem:quasi_geod} there exists $\epsilon >0$ and $A \geq 1$ such that for each $x \in \partial \Omega$  the curve $\sigma_{x}(t) = x+e^{-t} \epsilon n_x$ is an $(A,0)$-quasi-geodesic in $(\Omega, K_{\Omega})$. Let $M$ be as in the statement of Proposition~\ref{prop:gromov_basic} for $(A,0)$-quasi-geodesics.

Since $\partial \Omega$ is compact, there exists an $R>0$ such that
\begin{align*}
K_{\Omega}(\sigma_{x}(0), \sigma_{y}(0)) \leq R
\end{align*}
for all $x,y \in \partial \Omega$.

Now if $x,y \in \Uc$ then by Proposition~\ref{prop:C1} we have
\begin{align*}
\limsup_{t \rightarrow \infty} K_\Omega( \sigma_x(t), \sigma_y(t)) < \infty.
\end{align*}
So by Proposition~\ref{prop:gromov_basic}
\begin{align}
\label{eq:bd_dist}
\sup_{t \geq0} \Big(\max\left\{ K_{\Omega}(\sigma_{y}(t), \sigma_{x}),  K_{\Omega}(\sigma_{x}(t), \sigma_{y})\right\}\Big)  \leq M+2R.
\end{align}
But then the inequality in~\ref{eq:bd_dist} holds for all $x,y \in \overline{\Uc}$. But this contradicts Proposition~\ref{prop:asym_geom} when $x \in \Uc$ and $y \in \partial \Uc$.

\section{$L$-convexity and limits in the local Hausdorff topology}

When $L \geq 1$, Mercer~\cite{M1993} calls a convex set $\Omega$ \emph{$L$-convex} if there exists $C > 0$ such that
\begin{align*}
\delta_{\Omega}(p;v) \leq C\delta_{\Omega}(p)^{1/L}
\end{align*}
for all $p \in \Omega$ and $v \in \Cb^d$ non-zero. Every bounded strongly convex set is $2$-convex and Mercer~\cite[Section 3]{M1993} extended results about limits of complex geodesics in strongly convex sets (see~\cite[Section 2]{CHL1988}) to general $L$-convex sets. In~\cite{Z2014}, we extended some of these results to sequences of geodesic lines $\sigma_n : \Rb\rightarrow \Omega_n$ when $\Omega_n$ is a sequence of convex sets which converge in the local Hausdorff topology and satisfy a uniform $L$-convex property. In this section we recall some of these results.

Given a set $A \subset \Cb^d$, let $\Nc_{\epsilon}(A)$ denote the \emph{$\epsilon$-neighborhood of $A$} with respect to the Euclidean distance. The \emph{Hausdorff distance} between two bounded sets $A,B \subset \Cb^d$ is given by
\begin{align*}
d_{H}(A,B) = \inf \left\{ \epsilon >0 : A \subset \Nc_{\epsilon}(B) \text{ and } B \subset \Nc_{\epsilon}(A) \right\}.
\end{align*}
Equivalently,
\begin{align*}
d_H(A,B) = \max\left\{\sup_{a \in A}\inf_{b \in B} \norm{a-b}, \sup_{b \in B} \inf_{a \in A} \norm{a-b} \right\}.
\end{align*}
The Hausdorff distance induces a topology on the space of bounded open convex sets in $\Cb^d$.

The space of all convex sets in $\Cb^d$ can be given a topology from the local Hausdorff semi-norms. For $R >0$ and a set $A \subset \Cb^d$ let $A^{(R)} := A \cap B_R(0)$. Then define the \emph{local Hausdorff semi-norms} by
\begin{align*}
d_H^{(R)}(A,B) := d_H(A^{(R)}, B^{(R)}).
\end{align*}
A sequence of open convex sets $A_n$ is said to converge in the local Hausdorff topology to an open convex set $A$ if there exists some $R_0 \geq 0$ so that
\begin{align*}
\lim_{n \rightarrow \infty} d_H^{(R)}(A_n,A) = 0
\end{align*}
for all $R \geq R_0$.

Unsurprisingly, the Kobayashi distance is continuous with respect to the local Hausdorff topology.

\begin{theorem}\cite[Theorem 4.1]{Z2014}\label{thm:Kob_cont}
\label{thm:dist_conv}
Suppose $\Omega_n$ is a sequence of $\Cb$-proper convex open sets converging to a $\Cb$-proper convex open set $\Omega$ in the local Hausdorff topology. Then
\begin{align*}
K_{\Omega}(x,y) = \lim_{n \rightarrow \infty} K_{\Omega_n}(x,y)
\end{align*}
for all $x,y \in \Omega$ uniformly on compact sets of $\Omega \times \Omega$.
\end{theorem}

\begin{remark} Greene and Krantz~\cite{GK1982} also study the continuity of intrinsic metrics under various notions of convergence of domains.
\end{remark}

In the proof of Theorem~\ref{thm:main} we will be interested in the limit of a sequence of geodesics $\sigma_n : \Rb \rightarrow \Omega_n$ when the target domains converges in the local Hausdorff topology. As the next two Propositions show when the sequence $\Omega_n$ has uniform convexity properties these limits have nice properties.

\begin{proposition}\cite[Proposition 7.8]{Z2014}\label{prop:m_convex}
Suppose $\Omega_n$ is a sequence of $\Cb$-proper convex open sets converging to a $\Cb$-proper convex open set $\Omega$ in the local Hausdorff topology and there exists $C>0$, $L \geq 1$, and a compact set $K$ satisfying
\begin{align*}
\delta_{\Omega_n}(p;v) \leq C \delta_{\Omega_n}(p)^{1/L}
\end{align*}
for every $n$ sufficiently large, $p \in \Omega_n \cap K$, and $v \in \Cb^d$ non-zero.

If $\sigma_n : \Rb \rightarrow \Omega_n$ is a sequence of geodesics and there exists $a_n \leq b_n$ such that
\begin{enumerate}
\item $\sigma_n([a_n,b_n]) \subset K$,
\item $\lim_{n \rightarrow \infty} \norm{\sigma_n(a_n)-\sigma_n(b_n)} > 0$,
\end{enumerate}
then there exists $T_n \in [a_n,b_n]$ such that a subsequence of $t\rightarrow \sigma_n(t+T_n)$ converges locally uniformly to a geodesic $\sigma:\Rb \rightarrow \Omega$.
\end{proposition}

\begin{remark} Proposition 7.8 in~\cite{Z2014} assumes that $K=B_R(0)$ for some $R>0$. The proof taken verbatim implies the more general case stated here.
\end{remark}

\begin{proposition}\cite[Proposition 7.9]{Z2014}\label{prop:limits_exist}
Suppose $\Omega_n$ is a sequence of $\Cb$-proper convex open sets converging to a $\Cb$-proper convex open set $\Omega$ in the local Hausdorff topology and for any $R>r > 0$ there exists $C=C(R,r) >0$ and $L=L(R,r) \geq 1$ such that
\begin{align*}
\delta_{\Omega_n}(p; v) \leq C\delta_{\Omega_n}(p)^{1/L}
\end{align*}
for all $n$ sufficiently large, $p \in \Omega_n \cap \{ r \leq \norm{z} \leq R\}$, and $v \in \Cb^d$ non-zero.

Assume $\sigma_n : \Rb \rightarrow \Omega_n$ is a sequence of geodesics converging locally uniformly to a geodesic $\sigma:\Rb \rightarrow \Omega$. If $t_n \rightarrow \infty$ is a sequence such that $\lim_{n \rightarrow \infty} \sigma_n(t_n) = x_{\infty} \in \overline{\Cb^d}$,  then
\begin{align*}
\lim_{t \rightarrow \infty} \sigma(t) = x_{\infty}.
\end{align*}
\end{proposition}

\begin{remark} Proposition 7.9 in~\cite{Z2014} is slightly less general ($r$ is assumed to be zero), but the proof taken essentially verbatim implies the more general case stated here.
\end{remark}

\begin{proposition}\label{prop:finite_type}
Suppose that $F_n:\Cb^d \rightarrow \Rb_{\geq 0}$ is a sequence of continuous functions which are $C^\infty$ on $\Cb^d \setminus \{0\}$, non-negative, convex, and $F_n(0)=0$.  Assume that the $F_n$ converges locally uniformly to a function $F:\Cb^d \rightarrow \Rb_{\geq 0}$ in the $C^0$ topology on $\Cb^d$ and in $C^\infty$ topology on $\Cb^d \setminus \{0\}$.

If $\Omega_F$ is $\Cb$-proper and has finite type away from 0, then for any $R >  r > 0$ there exists $N=N(R,r) \geq 1$, $L=L(R,r) \geq 1$, and $C=C(R,r) > 0$ such that
\begin{align*}
\delta_{\Omega_{F_n}}(p;v) \leq C\delta_{\Omega_{F_n}}(p)^{1/L}
\end{align*}
for all $n \geq N$, $p \in \Omega_{F_n} \cap \{ r \leq \norm{z} \leq R\}$ and $v \in \Cb^{d+1}$ non-zero.
\end{proposition}

This is a special case of Proposition 9.3 and Example 9.4 in~\cite{Z2014}.

\section{Proof of Theorem~\ref{thm:main}}

Suppose $F:\Cb^d \rightarrow \Rb_{\geq 0}$ is a continuous function which is $C^\infty$ on $\Cb^d \setminus \{0\}$, non-negative, convex, and $F(0)=0$. As in the introduction let
\begin{align*}
\Omega_F =  \{ (z_0, \dots, z_d) \in \Cb^{d+1} : \Imaginary(z_0) > F(z_1, \dots, z_d) \}.
\end{align*}
Assume that $\Omega_F$ is $\Cb$-proper, has finite type away from 0, and $F$ is $(\delta_1,\dots, \delta_d)$-homogeneous

Suppose for a contradiction that  $(\Omega_F, K_{\Omega_F})$ is not Gromov hyperbolic. Then there exists points $x_n,y_n, z_n \in \Omega_F$, geodesic segments $\sigma_{x_ny_n}, \sigma_{y_nz_n}, \sigma_{z_nx_n}$ joining them, and a point $u_n$ in the image of $\sigma_{x_ny_n}$ such that
\begin{align*}
K_{\Omega_F}(u_n, \sigma_{y_nz_n} \cup \sigma_{z_nx_n}) > n.
\end{align*}
For $t > 0$, $\Omega_F$ is invariant under the holomorphic transformation
\begin{align*}
(z_0, z_1, \dots, z_d) \rightarrow \left( t z_0, t^{\delta_1} z_1, \dots, t^{\delta_d} z_d\right).
\end{align*}
So we may assume that $\norm{u_n}=1$. By passing to a subsequence we can also assume that $u_n \rightarrow u_\infty \in \overline{\Omega}_F$. Now there are two cases to consider $u_{\infty} \in \partial \Omega_F$ and $u_{\infty} \in \Omega_F$.

\subsection{Case 1:} Suppose that $u_{\infty} \in \partial \Omega_F$. Since $\norm{u_{\infty}}=1$ we see that $\partial \Omega_F$ has finite line type at $u_\infty$.

We will need two results from~\cite{Z2014}:

\begin{theorem}\label{thm:gaussier}\cite[Theorem 10.1]{Z2014}
Suppose $\Omega \subset \Cb^{d+1}$ is a convex open set such that $\partial \Omega$ is $C^L$ and has finite line type $L$ near some $\xi \in \partial \Omega$. If $q_n \in \Omega$ is a sequence converging to $\xi$, then there exists $n_k \rightarrow \infty$ and affine maps $A_k \in \Aff(\Cb^d)$ such that
\begin{enumerate}
\item $A_k \Omega$ converges in the local Hausdorff topology to a $\Cb$-proper convex open set $\wh{\Omega}$ of the form:
\begin{align*}
\wh{\Omega} = \{ (z_0,z_1 \dots, z_d) \in \Cb^{d} : \Real(z_0)  > P(z_1,z_2, \dots, z_d) \}
\end{align*}
where $P$ is a non-negative non-degenerate convex polynomial with $P(0)=0$,
\item $A_k u_{n_k} \rightarrow u_{\infty} \in \wh{\Omega}$, and
\item for any $R>0$ there exists $C=C(R)>0$ and $N=N(R)>0$ such that
\begin{align*}
\delta_{\Omega_n}(p; v) \leq C \delta_{\Omega_n}(p)^{1/L}
\end{align*}
for all $n>N$, $p \in B_R(0) \cap \Omega_n$, and $v \in \Cb^d$ non-zero.
\end{enumerate}
\end{theorem}

\begin{remark}
With the exception of part (3) the above theorem follows from an argument of Gaussier~\cite{G1997}.
\end{remark}

\begin{proposition}\label{prop:geodesics_well_behaved}\cite[Proposition 12.2]{Z2014}
Suppose $\Omega$ is a domain of the form
\begin{align*}
\Omega = \{ (z_0, \dots, z_d) \in \Cb^{d+1} : \Real(z_0)  > P(z_1, \dots, z_d) \}
\end{align*}
where $P$ is a non-negative non-degenerate convex polynomial with $P(0)=0$. If $\sigma: \Rb \rightarrow \Omega$ is a geodesic, then $\lim_{t \rightarrow -\infty} \sigma(t)$ and $\lim_{t \rightarrow \infty} \sigma(t)$ both exist in $\overline{\Cb^d}$ and are distinct.
\end{proposition}

We can now complete the proof of Theorem~\ref{thm:main} in case 1. By passing to a subsequence there exists affine maps $A_n \in \Aff(\Cb^d)$ and a $\Cb$-proper convex open set $\wh{\Omega}$ such that

\begin{enumerate}
\item $\Omega_n: = A_n \Omega_F \rightarrow \wh{\Omega}$ in the local Hausdorff topology,
\item $A_n u_n \rightarrow u_{\infty} \in \wh{\Omega}$,
\item for any $R>0$ there exists $C, N, L>0$ such that
\begin{align*}
\delta_{\Omega_n}(p; v) \leq C \delta_{\Omega_n}(p)^{1/L}
\end{align*}
for all $n>N$, $p \in B_R(0) \cap \Omega_n$, and $v \in \Cb^d$ non-zero,
\item if $\sigma: \Rb \rightarrow \wh{\Omega}$ is a geodesic then $\lim_{t \rightarrow -\infty} \sigma(t)$ and $\lim_{t \rightarrow \infty} \sigma(t)$ both exist in $\overline{\Cb^d}$ and are distinct.
\end{enumerate}
By passing to another subsequence we can suppose that $A_n x_n \rightarrow x_{\infty}$, $A_n y_n \rightarrow y_{\infty}$, and $A_n z_n \rightarrow z_{\infty}$ for some $x_{\infty}, y_{\infty}, z_{\infty} \in \overline{\Cb^d} = \Cb^d \cup \{\infty\}$.

Parametrize $\sigma_{x_ny_n}$ such that $\sigma_{x_ny_n}(0)=u_n$ then using Theorem~\ref{thm:Kob_cont} we can pass to a subsequence such that $A_n\sigma_{x_ny_n}$ converges locally uniformly to a geodesic $\sigma: \Rb \rightarrow \wh{\Omega}$. Moreover, by Proposition~\ref{prop:limits_exist}
\begin{align*}
\lim_{t \rightarrow -\infty} \sigma(t) = x_{\infty} \text{ and } \lim_{t \rightarrow +\infty} \sigma(t) = y_{\infty}.
\end{align*}
Then we must have that $x_{\infty} \neq y_{\infty}$.

So $z_{\infty}$ does not equal at least one of $x_{\infty}$ or $y_{\infty}$. By relabeling we can suppose that $x_{\infty} \neq z_{\infty}$. Since $x_{\infty} \neq z_{\infty}$ at least one is finite and hence by Proposition~\ref{prop:m_convex} we may pass to a subsequence and parametrize $A_n\sigma_{x_nz_n}$ so that it converges locally uniformly to a geodesic $\wh{\sigma}:\Rb \rightarrow \wh{\Omega}$. But then
\begin{align*}
K_{\wh{\Omega}}(u_{\infty},\wh{\sigma}(0))
&= \lim_{n \rightarrow \infty} K_{\Omega_n}(A_nu_n, A_n\sigma_{x_n y_n} (0))\\
&= \lim_{n \rightarrow \infty} K_{\Omega_F}(u_n, \sigma_{x_n y_n} (0))\\
& \geq  \lim_{n \rightarrow \infty} K_{\Omega_F}(u_n, \sigma_{x_nz_n})=\infty
\end{align*}
which is a contradiction.

\subsection{Case 2:} Suppose that $u_{\infty} \in \Omega_F$. By passing to a subsequence we can suppose that $x_n \rightarrow x_{\infty}$, $y_n \rightarrow y_{\infty}$, and $z_n \rightarrow z_{\infty}$ for some $x_{\infty}, y_{\infty}, z_{\infty} \in \overline{\Cb^{d+1}} = \Cb^{d+1} \cup \{\infty\}$. Since
\begin{align*}
K_{\Omega_F}(u_n, \{x_n,y_n,z_n\}) > n
\end{align*}
we must have that $x_{\infty}, y_{\infty}, z_{\infty} \in \partial \Omega_F \cup \{\infty\}$.

Fix $R > r > 0$ then by Proposition~\ref{prop:finite_type} there exists $C >0$ and $L \geq 1$ such that
\begin{align*}
\delta_{\Omega_F}(p; v) \leq C \delta_{\Omega_F}(p)^{1/L}
\end{align*}
for all $p \in \Omega_F \cap \{ r \leq \norm{z} \leq R\}$ and $v \in \Cb^{d+1}$ non-zero. In particular we can apply Proposition~\ref{prop:m_convex} and Proposition~\ref{prop:limits_exist} when taking the limits of geodesics $\sigma_n : \Rb \rightarrow \Omega_F$.

Now parametrize $\sigma_{x_ny_n}$ such that $\sigma_{x_ny_n}(0)=u_n$ then using the Arzel{\`a}-Ascoli theorem we can pass to a subsequence such that $\sigma_{x_ny_n}$ converges locally uniformly to a geodesic $\sigma: \Rb \rightarrow \Omega_F$. Then by Proposition~\ref{prop:limits_exist}
\begin{align*}
\lim_{t \rightarrow -\infty} \sigma(t) = x_{\infty} \text{ and } \lim_{t \rightarrow +\infty} \sigma(t) = y_{\infty}.
\end{align*}

We now claim:

\begin{proposition}\label{prop:well_behaved}
If $\gamma: \Rb \rightarrow \Omega_F$ is a geodesic, then $\lim_{t \rightarrow -\infty} \gamma(t)$ and $\lim_{t \rightarrow \infty} \gamma(t)$ both exist in $\overline{\Cb^{d+1}}$ and are distinct.
\end{proposition}

Delaying the proof of Proposition~\ref{prop:well_behaved} we can complete the proof of Theorem~\ref{thm:main}. Now $x_{\infty} \neq y_{\infty}$, so $z_{\infty}$ does not equal at least one of $x_{\infty}$ or $y_{\infty}$. By relabeling we can suppose that $x_{\infty} \neq z_{\infty}$. Since $x_{\infty} \neq z_{\infty}$ at least one is finite and hence by Proposition~\ref{prop:m_convex} we may pass to a subsequence and parametrize $\sigma_{x_nz_n}$ so that it converges locally uniformly to a geodesic $\wh{\sigma}:\Rb \rightarrow \Omega_F$. But then
\begin{align*}
K_{\Omega_F}(u_{\infty},\wh{\sigma}(0))
&= \lim_{n \rightarrow \infty} K_{\Omega_F}(u_n, \sigma_{x_n y_n} (0))\\
& \geq  \lim_{n \rightarrow \infty} K_{\Omega_F}(u_n, \sigma_{x_nz_n})=\infty
\end{align*}
which is a contradiction.

To finish the argument we now prove Proposition~\ref{prop:well_behaved} which will require one result from~\cite{Z2014}. Define the \emph{Gromov product} on $\Omega$ by
\begin{align*}
(p|q)_o^{\Omega} = \frac{1}{2} \Big(K_{\Omega}(p,o)+K_{\Omega}(o,q)-K_{\Omega}(p,q) \Big).
\end{align*}

\begin{proposition}\cite[Proposition 11.3]{Z2014}\label{prop:gromov_prod_finite}
Suppose $\Omega$ is a $\Cb$-proper convex open set. Assume $p_n, q_n \in \Omega$ are sequences with $\lim_{n \rightarrow \infty} p_n = \xi^+ \in \partial \Omega$, $\lim_{n \rightarrow \infty} q_n = \xi^- \in \partial \Omega \cup \{ \infty\}$, and
\begin{align*}
\liminf_{n,m \rightarrow \infty} \ (p_n | q_m)_o^{\Omega} < \infty.
\end{align*}
If $\partial \Omega$ is $C^2$ near $\xi^+$, then $\xi^+ \neq \xi^-$.
\end{proposition}

\begin{proof}[Proof of Proposition~\ref{prop:well_behaved}]
First suppose that $\lim_{t \rightarrow \infty} \gamma(t)$ does not exist. Then there exists $s_n,t_n \rightarrow \infty$ and $\xi_1, \xi_2 \in \Cb^{d+1} \cup \{\infty\}$ such that
\begin{align*}
\lim_{n \rightarrow \infty} \gamma(s_n) = \xi_1 \neq \xi_2 = \lim_{n \rightarrow \infty} \gamma(t_n).
\end{align*}
Now  (up to relabeling) either $\xi_1 =0$ and $\xi_2 = \infty$ or $\xi_1 \in \partial \Omega_F \setminus \{0\}$. In either case there exists $R > r > 0$ and $[a_n, b_n] \subset [ \min\{s_n,t_n\}, \infty)$ such that
\begin{enumerate}
\item $\gamma([a_n,b_n]) \subset \{ r < \norm{z} < R\}$ for large $n$,
\item $\lim_{n \rightarrow \infty} \norm{\gamma(a_n)-\gamma(b_n)} > 0$.
\end{enumerate}
So by Proposition~\ref{prop:m_convex} we can pass to a subsequence and find $T_n \in [a_n, b_n]$ such that $t \rightarrow \sigma(t+T_n)$ converges locally uniformly to a geodesic $\wh{\gamma}:\Rb \rightarrow \Omega_F$. But then
\begin{align*}
K_{\Omega_F}(\gamma(0), \wh{\gamma}(0)) = \lim_{n \rightarrow\infty} K_{\Omega_F}(\gamma(0), \gamma(T_n)) \geq \limsup_{n \rightarrow \infty} a_n \geq
 \lim_{n \rightarrow \infty} \min\{t_n, s_n\} = \infty
\end{align*}
which is a contradiction. Thus the limits $\lim_{t \rightarrow -\infty} \gamma(t)$ and $\lim_{t \rightarrow \infty} \gamma(t)$ both exist.

Now suppose for a contradiction that $\lim_{t \rightarrow -\infty} \gamma(t) = \xi = \lim_{t \rightarrow \infty} \gamma(t)$. Notice that
\begin{align*}
\lim_{t \rightarrow \infty} (\gamma(t)|\gamma(-t))_{\gamma(0)}^{\Omega} = \lim_{t \rightarrow \infty} 0 = 0,
\end{align*}
then since $\partial \Omega_F \setminus \{0\}$ is $C^\infty$ Proposition~\ref{prop:gromov_prod_finite} implies that either $\xi=0$ or $\xi=\infty$.

If $\xi=0$ consider the geodesics
\begin{align*}
\gamma_n(t):=
\begin{pmatrix} n & & & \\
& n^{\delta_1} & & \\
& & \ddots & \\
& & & n^{\delta_d}
\end{pmatrix}\gamma(t).
\end{align*}
Notice that
\begin{align*}
\lim_{t \rightarrow -\infty} \gamma_n(t) = 0 = \lim_{t \rightarrow \infty} \gamma_n(t)
\end{align*}
for any $n$, but $\lim_{n \rightarrow \infty} \gamma_n(0) = \infty$. Thus by Proposition~\ref{prop:m_convex} we can pass to a subsequence $n_k \rightarrow \infty$ and find $\alpha_k \in (-\infty, 0]$ and $\beta_k \in [0,\infty)$ such that the geodesics $t \rightarrow \gamma_{n_k}(t+\alpha_k)$ and $t \rightarrow \gamma_{n_k}(t+\beta_k)$ converge locally uniformly to geodesics $\wh{\gamma}_1, \wh{\gamma}_2 : \Rb \rightarrow \Omega_F$. Since $\gamma_n(0) \rightarrow \infty$ and $0 \in \partial \Omega_F$ Lemma~\ref{lem:convex_lower_bd_2} implies that $\alpha_k \rightarrow -\infty$ and $\beta_k \rightarrow \infty$. Then
\begin{align*}
K_{\Omega_F}(\wh{\gamma}_1(0), \wh{\gamma}_2(0)) = \lim_{k \rightarrow \infty} K_{\Omega_F}(\gamma_{n_k}(\alpha_k), \gamma_{n_k}(\beta_k)) = \lim_{k \rightarrow \infty} \beta_k - \alpha_k = \infty
\end{align*}
which is a contradiction.

The case in which $\xi=\infty$ is shown to be impossible in a similar fashion by considering the geodesics \begin{align*}
\gamma_n(t):=
\begin{pmatrix} n^{-1} & & & \\
& n^{-\delta_1} & & \\
& & \ddots & \\
& & & n^{-\delta_d}
\end{pmatrix}\gamma(t).
\end{align*}
Thus $\lim_{t \rightarrow -\infty} \gamma(t)$ and $\lim_{t \rightarrow \infty} \gamma(t)$ both exist in $\overline{\Cb^{d+1}}$ and are distinct.
\end{proof}

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