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\begin{document}

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\section*{ball}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{topology}{}\paragraph*{{Topology}}\label{topology}

[[!include topology - contents]]

\hypertarget{differential_geometry}{}\paragraph*{{Differential geometry}}\label{differential_geometry}

[[!include synthetic differential geometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{geometric}{Geometric}\dotfill \pageref*{geometric} \linebreak
\noindent\hyperlink{combinatorial}{Combinatorial}\dotfill \pageref*{combinatorial} \linebreak
\noindent\hyperlink{Properties}{Properties}\dotfill \pageref*{Properties} \linebreak
\noindent\hyperlink{closed_balls}{Closed balls}\dotfill \pageref*{closed_balls} \linebreak
\noindent\hyperlink{open_balls}{Open Balls}\dotfill \pageref*{open_balls} \linebreak
\noindent\hyperlink{good_covers_by_balls}{Good covers by balls}\dotfill \pageref*{good_covers_by_balls} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak
\noindent\hyperlink{geometric_2}{Geometric}\dotfill \pageref*{geometric_2} \linebreak
\noindent\hyperlink{ReferencesStarShapedReasonDiffeomorphicToOpenBall}{Star-shaped regions diffeomorphic to open ball}\dotfill \pageref*{ReferencesStarShapedReasonDiffeomorphicToOpenBall} \linebreak
\noindent\hyperlink{combinatorial_2}{Combinatorial}\dotfill \pageref*{combinatorial_2} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{geometric}{}\subsubsection*{{Geometric}}\label{geometric}

For $n \in \mathbb{N}$ a [[natural number]], the $n$-[[dimension|dimensional]] \textbf{ball} or \textbf{$n$-disk} in $\mathbb{R}^n$ is the [[topological space]]

\begin{displaymath}
D^n := \{ \vec x \in \mathbb{R}^n | \sum_{i} (x^i)^2 \leq 1\}
  \subset \mathbb{R}^n
\end{displaymath}
equipped with the [[induced topology]] as a subspace of the [[Cartesian space]] $\mathbb{R}^n$.

Its [[interior]] is the \textbf{open $n$-ball}

\begin{displaymath}
\mathbb{B}^n := \{ \vec x \in \mathbb{R}^n | \sum_{i} (x^i)^2 \lt 1 \}
  \subset \mathbb{R}^n
  \,.
\end{displaymath}
Its [[boundary]] is the $(n-1)$-[[sphere]].

More generally, for $(X,d)$ a [[metric space]] then an open ball in $X$ is a subset of the form

\begin{displaymath}
B(x,r) \coloneqq \{x \in X \;|\; d(x,y) \lt r \}
\end{displaymath}
for $x \in X$ and $r \in (0,\infty) \subset \mathbb{R}$. (The collection of all open balls in $X$ form the [[basis of a topology|basis]] of the [[metric topology]] on $X$.)

\hypertarget{combinatorial}{}\subsubsection*{{Combinatorial}}\label{combinatorial}

There are also combinatorial notions of \emph{disks}. For instance that due to (\hyperlink{Joyal}{Joyal}), as entering the definition of the [[Theta-category]]. See for instance (\hyperlink{MakkaiZawadowski}{Makkai-Zawadowski}).

\hypertarget{Properties}{}\subsection*{{Properties}}\label{Properties}

\hypertarget{closed_balls}{}\subsubsection*{{Closed balls}}\label{closed_balls}

A simple result on the \emph{homeomorphism} type of \emph{closed} balls is the following:

\begin{theorem}
\label{}\hypertarget{}{}
A [[compact space|compact]] [[convex subset|convex]] [[subset]] $D$ in $\mathbb{R}^n$ with [[nonempty set|nonempty]] [[interior]] is [[homeomorphic]] to $D^n$.

\end{theorem}
\begin{proof}
Without loss of generality we may suppose the origin is an interior point of $D$. We claim that the map $\phi: v \mapsto v/{\|v\|}$ maps the boundary $\partial D$ homeomorphically onto $S^{n-1}$. By convexity, $D$ is homeomorphic to the cone on $\partial D$, and therefore to the cone on $S^{n-1}$ which is $D^n$.

The claim reduces to the following three steps.

\begin{enumerate}%
\item The restricted map $\phi: \partial D \to S^{n-1}$ is continuous.


\item It's surjective: $D$ contains a ball $B = B_{\varepsilon}(0)$ in its interior, and for each $x \in B$, the positive ray through $x$ intersects $D$ in a bounded half-open line segment. For the extreme point $v$ on this line segment, $\phi(v) = \phi(x)$. Thus every unit vector $u \in S^{n-1}$ is of the form $\phi(v)$ for some extreme point $v \in D$, and such extreme points lie in $\partial D$.


\item It's injective: for this we need to show that if $v, w \in \partial D$ are distinct points, then neither is a positive multiple of the other. Supposing otherwise, we have $w = t v$ for $t \gt 1$, say. Let $B$ be a ball inside $D$ containing $0$; then the convex hull of $\{w\} \cup B$ is contained in $D$ and contains $v$ as an interior point, contradiction.



\end{enumerate}
So the unit vector map, being a continuous bijection $\partial D \to S^{n-1}$ between [[compact Hausdorff space]]s, is a homeomorphism.

\end{proof}
\begin{cor}
\label{}\hypertarget{}{}
Any compact convex set $D$ of $\mathbb{R}^n$ is homeomorphic to a disk.

\end{cor}
\begin{proof}
$D$ has nonempty interior relative to its affine span which is some $k$-plane, and then $D$ is homeomorphic to $D^k$ by the theorem.

\end{proof}
\hypertarget{open_balls}{}\subsubsection*{{Open Balls}}\label{open_balls}

Open balls are a little less rigid than closed balls, in that one can more easily manipulate them within the \emph{smooth} category:

\begin{lemma}
\label{}\hypertarget{}{}
The open $n$-ball is [[homeomorphic]] and even [[diffeomorphic]] to the [[Cartesian space]] $\mathbb{R}^n$

\begin{displaymath}
\mathbb{B}^n \simeq \mathbb{R}^n
  \,.
\end{displaymath}
\end{lemma}
\begin{proof}
For instance, the smooth map

\begin{displaymath}
x\mapsto \frac{x}{\sqrt{1+|x|^2}} : \mathbb{R}^n \to \mathbb{B}^n
\end{displaymath}
has smooth inverse

\begin{displaymath}
y\mapsto \frac{y}{\sqrt{1-|y|^2}} : \mathbb{B}^n \to \mathbb{R}^n.
\end{displaymath}
\end{proof}
This probe from ${\mathbb{R}}^n$ witnesses the property that the open $n$-ball is a ([[smooth manifold|smooth]]) [[manifold]]. Hence, each (smooth) $n$-dimensional manifold is locally isomorphic to both ${\mathbb{R}}^n$ and $\mathbb{B}^n$.

From general existence results about [[smooth structure]]s on [[Cartesian space]]s we have that

\begin{theorem}
\label{}\hypertarget{}{}
In [[dimension]] $d \in \mathbb{N}$ for $d \neq 4$ we have:

every open subset of $\mathbb{R}^d$ which is [[homeomorphic]] to $\mathbb{B}^d$ is also [[diffeomorphic]] to it.

\end{theorem}
See the first page of (\hyperlink{Ozols}{Ozols}) for a list of references.

\begin{remark}
\label{}\hypertarget{}{}
In dimension 4 the analog statement fails due to the existence of [[exotic smooth structure]]s on $\mathbb{R}^4$. See \hyperlink{DeMFreed}{De Michelis-Freedman}.

\end{remark}
\begin{theorem}
\label{StarShapedOpenDiffeomorphicToOpenBall}\hypertarget{StarShapedOpenDiffeomorphicToOpenBall}{}
\textbf{(star-shaped domains are diffeomorphic to open balls)}

Let $C \subset \mathbb{R}^n$ be a [[star-shaped]] [[open subset]] of a [[Cartesian space]]. Then $C$ is [[diffeomorphic]] to $\mathbb{R}^n$.

\end{theorem}
\begin{remark}
\label{LiteratureOnStarShapedOpenDiffeoToOpenBall}\hypertarget{LiteratureOnStarShapedOpenDiffeoToOpenBall}{}
Theorem \ref{StarShapedOpenDiffeomorphicToOpenBall} is a [[folk theorem]], but explicit \textbf{proofs} in the literature are hard to find. See the discussion at \hyperlink{References}{References}. An explicit proof has been written out by Stefan Born, and this appears as the proof of \href{http://www.math.tu-berlin.de/~ferus/ANA/Ana3.pdf#page=154}{theorem 237} in (\hyperlink{Ferus07}{Ferus 07}). A simplr proof is given in \hyperlink{GonnordTosel98}{Gonnord-Tosel 98} reproduced \href{http://mathoverflow.net/a/212595/381}{here}.

\end{remark}
Here is another proof:

\begin{proof}
Suppose $T$ is a star-shaped open subset of ${\mathbb {R}}^n$ centered at the origin. Theorem 2.29 in \hyperlink{Lee}{Lee} proves that there is a function $f$ on ${\mathbb{R}}^n$ such that $f\gt 0$ on $T$ and $f$ vanishes on the complement of $T$. By applying [[bump functions]] we can assume that $f\le 1$ everywhere and $f=1$ in an open $\epsilon$-neighborhood of the origin; by rescaling the ambient space we can assume $\epsilon=2$.

The smooth vector field $V\colon x\mapsto f(x)\cdot x/{\|x\|}$ is defined on the complement of the origin in $T$. Multiply $V$ by a smooth bump function $0\le b\le 1$ such that $b=1$ for ${\|x\|} \gt 1/2$ and $b=0$ in a neighborhood of 0. The new vector field $V$ extends smoothly to the origin and defines a smooth global flow $F\colon \mathbb{R} \times T\to T$. (The parameter of the flow is all of $\mathbb{R}$ and not just some interval $(-\infty,A)$ because the norm of $V$ is bounded by 1.) Observe that for $1/2\lt {\|x\|} \lt 2$ the vector field $V$ equals $x\mapsto x/{\|x\|}$. Also, all flow lines of $V$ are radial rays.

Now define the flow map $p\colon{\mathbb{R}}^n_{\gt 1/2}\to T_{\gt 1/2}$ as $x\mapsto F({\|x\|}-1, \frac{x}{{\|x\|}})$ for ${\|x\|} \gt 1/2$. (The subscript $\gt 1/2$ removes the closed ball of radius $1/2$.) The flow map is the composition of two diffeomorphisms,

\begin{displaymath}
{\mathbb{R}}^n_{\gt 1/2}\to(-1/2,\infty)\times S^{n-1} \to T_{\gt 1/2},
\end{displaymath}
hence itself is a diffeomorphism. (Note particularly that the latter map is surjective. In detail: a flow line is a smooth map of the form $L: (A,B) \to T$, where $A$ and $B$ can be finite or infinite. If $B$ is finite and the limit of $L(t)$ as $t \to B$ exists, then the vector field $V$ vanishes at $B$. In our case $V$ can only vanish at the boundary of $T$, which is precisely what we want for surjectivity.)

Finally, define the desired diffeomorphism $d\colon{\mathbb{R}}^n\to T$ as the gluing of the identity map for ${\|x\|} \lt 2$ and as $p$ for ${\|x\|}\gt 1/2$. The map $g$ is smooth because for $1/2\lt {\|x\|} \lt 2$ both definitions give the same value.

\end{proof}
\begin{example}
\label{}\hypertarget{}{}
Let $I(\Delta^n) \subset \mathbb{R}^n$ be the [[interior]] of the standard $n$-[[simplex]]. Then there is a diffeomorphism to $\mathbb{B}^n$ defined as follows:

Parameterize the $n$-simplex as

\begin{displaymath}
I(\Delta^n) 
  =
  \left\{
    (x^1, \cdots, x^n) \in \mathbb{R}
    |
     (\forall i : x^i \gt 0)\; and \; ( \sum_{i=1}^n x^i \lt 1)
  \right\}
  \,.
\end{displaymath}
Then define the map $f : I(\Delta^n) \to \mathbb{R}^n$ by

\begin{displaymath}
(x^1, \ldots, x^n) 
  \mapsto 
  (\log(\frac{x^1}{1 - x^1 - \ldots -x^n}), 
  \ldots, 
  \log(\frac{x^n}{1 - x^1 - \ldots - x^n}))
  \,.
\end{displaymath}
\end{example}
(Thanks to [[Todd Trimble]].) One way to think about it is that $I(\Delta^n)$ is the positive orthant of an open $n$-ball in $l^1$ norm, so that in the opposite direction we have a chain of invertible maps

\begin{displaymath}
\itexarray{
\mathbb{R}^n & \stackrel{\exp^n}{\to} & \mathbb{R}_+^n & \to & I(\Delta^n) \\ 
 & & \vec{x} & \mapsto & \vec{x}/(1 + {\|\vec{x}\|}_1) 
}
\end{displaymath}
which we simply invert to get the map $f$ above.

\hypertarget{good_covers_by_balls}{}\subsubsection*{{Good covers by balls}}\label{good_covers_by_balls}

One central application of balls is as building blocks for [[covering]]s. See [[good open cover]] for some statements.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[unit ball]]

\end{itemize}
\hypertarget{References}{}\subsection*{{References}}\label{References}

\hypertarget{geometric_2}{}\subsubsection*{{Geometric}}\label{geometric_2}

\begin{itemize}%
\item V. Ozols, \emph{Largest normal neighbourhoods} , Proceedings of the American Mathematical Society Vol. 61, No. 1 (Nov., 1976), pp. 99-101 (\href{http://www.jstor.org/stable/2041672}{jstor})

\end{itemize}
That an open subset $U \subseteq \mathbb{R}^4$ homeomorphic to $\mathbb{R}^4$ equipped with the smooth structure inherited as an open submanifold of $\mathbb{R}^4$ might nevertheless be non-diffeomorphic to $\mathbb{R}^4$, see

\begin{itemize}%
\item De Michelis, Stefano; Freedman, Michael H. (1992) ``Uncountably many exotic $\mathbb{R}^4$`s in standard 4-space'', J. Diff. Geom. 35, pp. 219-254.

\end{itemize}
\hypertarget{ReferencesStarShapedReasonDiffeomorphicToOpenBall}{}\subsubsection*{{Star-shaped regions diffeomorphic to open ball}}\label{ReferencesStarShapedReasonDiffeomorphicToOpenBall}

The proof that open star-shaped regions are diffeomorphic to a ball appears as \href{http://www.math.tu-berlin.de/~ferus/ANA/Ana3.pdf#page=154}{theorem 237} in

\begin{itemize}%
\item [[Dirk Ferus]], \emph{Analysis III} (2007) (\href{http://www.math.tu-berlin.de/~ferus/ANA/Ana3.pdf}{pdf})

\end{itemize}
It is a lengthy proof, due to Stefan Born.

A simpler version of the proof apparently appears on page 60 of

\begin{itemize}%
\item St\'e{}phane Gonnord, Nicolas Tosel, \emph{Calcul Diff\'e{}rentiel}, ellipses (1998)

\end{itemize}
and is reproduced in

\begin{itemize}%
\item MO \href{http://mathoverflow.net/a/212595/381}{here}

\end{itemize}
Apparently this proof is little known. For instance in a remark below lemma 10.5.5 of

\begin{itemize}%
\item Lawrence Conlon, \emph{Differentiable manifolds}, Birkh\"a{}user (last edition 2008)

\end{itemize}
it says:

\begin{quote}%
It seems that open star shaped sets $U \subset M$ are always diffeomorphic to $\mathbb{R}^n$, but this is extremely difficult to prove.


\end{quote}
And in

\begin{itemize}%
\item Jeffrey Lee, \emph{Manifolds and differential geometry} (2009)

\end{itemize}
one finds the statement:

\begin{quote}%
Actually, the assertion that an open geodesically convex set in a Riemannian manifold is diffeomorphic to $\mathbb{R}^n$ is common in literature, but it is a more subtle issue than it may seem, and references to a complete proof are hard to find (but see Grom).


\end{quote}
Here ``Grom'' refers to

\begin{itemize}%
\item [[Mikhail Gromov]], \emph{Convex sets and K\"a{}hler manifolds}, Advances in differential geometry and topology. F. Tricerri ed., World Sci., Singapore, (1990), 1-38. (\href{http://www.ihes.fr/~gromov/PDF/%5B68%5D.pdf}{pdf})

\end{itemize}
where the relevant statement is 1.4.C1 on page 8. Note however that the diffeomorphism considered there is only of $C^1$ class, not $C^\infty$, so this is not a proof either.

For a discussion of diffeomorphisms between geodesically convex regions and open balls see at \emph{[[good open cover]]}.

See also the Math Overflow discussion \href{http://mathoverflow.net/questions/41853/explicit-diffeomorphim-between-open-simplex-and-open-ball}{here}.

\hypertarget{combinatorial_2}{}\subsubsection*{{Combinatorial}}\label{combinatorial_2}

\begin{itemize}%
\item [[Andre Joyal]], \emph{Disks, duality and Theta-categories} ([[JoyalThetaCategories.pdf:file]])


\item [[Mihaly Makkai]], Marek Zawadowski, \emph{Duality for Simple $\omega$-Categories and Disks} (\href{http://www.emis.de/journals/TAC/volumes/8/n7/8-07abs.html}{TAC})



\end{itemize}
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