nLab ball

Definition

Geometric

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

$$D^n := \{ \vec x \in \mathbb{R}^n | \sum_{i} (x^i)^2 \leq 1\} \subset \mathbb{R}^n$$

equipped with the induced topology as a subspace of the Cartesian space $\mathbb{R}^n$.

Its interior is the open $n$-ball

$$\mathbb{B}^n := \{ \vec x \in \mathbb{R}^n | \sum_{i} (x^i)^2 \lt 1 \} \subset \mathbb{R}^n \,.$$

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

$$B(x,r) \coloneqq \{y \in X \;|\; d(x,y) \lt r \}$$

for $x \in X$ and $r \in (0,\infty) \subset \mathbb{R}$. (The collection of all open balls in $X$ form the basis of the metric topology on $X$.)

Combinatorial

There are also combinatorial notions of disks. For instance that due to (Joyal), as entering the definition of the Theta-category. See for instance (Makkai-Zawadowski).

Properties

Closed balls

A simple result on the homeomorphism type of closed balls is the following:

Theorem

A compact convex subset $D$ in $\mathbb{R}^n$ with nonempty interior is homeomorphic to $D^n$.

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.

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

2. 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 $\partial D$.

3. 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.

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

Corollary

Any compact convex set $D$ of $\mathbb{R}^n$ is homeomorphic to a disk.

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.

Open Balls

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

Observation

The open $n$-ball is homeomorphic and even diffeomorphic to the Cartesian space $\mathbb{R}^n$

$$\mathbb{B}^n \simeq \mathbb{R}^n \,.$$

Proof

For instance, the smooth map

$$x\mapsto \frac{x}{\sqrt{1+|x|^2}} : \mathbb{R}^n \to \mathbb{B}^n$$

has smooth inverse

$$y\mapsto \frac{y}{\sqrt{1-|y|^2}} : \mathbb{B}^n \to \mathbb{R}^n.$$

This probe from ${\mathbb{R}}^n$ witnesses the property that the open $n$-ball is a (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 structures on Cartesian spaces we have that

Theorem

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.

See the first page of (Ozols) for a list of references.

Remark

In dimension 4 the analog statement fails due to the existence of exotic smooth structures on $\mathbb{R}^4$. See De Michelis-Freedman.

Theorem

(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 $\mathbb{R}^n$.

Remark

Theorem is a folk theorem, but explicit proofs in the literature are hard to find. See the discussion in the References-section here. An explicit proof has been written out by Stefan Born, and this appears as the proof of theorem 237 in (Ferus 07). A simpler proof is given in Gonnord-Tosel 98 reproduced here.

Here is another proof:

Proof

Suppose $T$ is a star-shaped open subset of ${\mathbb {R}}^n$ centered at the origin. Theorem 2.29 in Lee 2009 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,

$${\mathbb{R}}^n_{\gt 1/2}\to(-1/2,\infty)\times S^{n-1} \to T_{\gt 1/2},$$

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.

And here is another proof, due to Gonnord and Tosel, translated into English by Erwann Aubry and available on MathOverflow:

Theorem

Every open star-shaped set $\Omega$ in $\mathbb{R}^n$ is $C^\infty$-diffeomorphic to $\mathbb{R}^n$.

Proof

For convenience assume that $\Omega$ is star-shaped at $0$.

Let $F=\mathbf{R}^n\setminus\Omega$ and $\phi:\mathbf{R}^n\rightarrow\mathbb{R}_+$ (here $\mathbf{R}_+=[0,\infty)$) be a $C^\infty$-function such that $F=\phi^{-1}(\{0\})$. (Such $\phi$ exists by the Whitney extension theorem.)

Now we define $f:\Omega\rightarrow\mathbb{R}^n$ via the formula:

$$f(x)=\overbrace{\left[1+\left(\int_0^1\frac{dv}{\phi(vx)}\right)^2\|x\|^2\right]}^{\lambda(x)}\cdot x=\left[1+\left(\int_0^{\|x\|}\frac{dt}{\phi(t\frac{x}{\|x\|})}\right)^2\right]\cdot x.$$

Clearly $f$ is smooth on $\Omega$.

We set $A(x)=\sup\{t\gt0\mid t\frac{x}{\|x\|}\in\Omega\}$. $f$ sends injectively the segment (or ray) $[0,A(x))\frac{x}{\|x\|}$ to the ray $\mathbf{R}_+\frac{x}{\|x\|}$. Moreover, $f(0\frac{x}{\|X\|})=0$ and

$$\lim_{r\rightarrow A(x)}\left\|f(r\frac{x}{\|x\|})\right\|=\lim_{r\to A(x)}\left[1+\left(\int_0^{r}\frac{dt}{\phi\left(t\cdot\frac{rx}{\|x\|}\cdot\left\|\frac{\|x\|}{rx}\right\|\right)}\right)^2\right]\cdot r= \left[1+\left(\int_0^{A(x)}\frac{dt}{\phi(t\frac{x}{\|x\|})}\right)^2\right]\cdot A(x)=+\infty.$$

Indeed, if $A(x)=+\infty$, then it holds for obvious reason. If $A(x)\lt+\infty$, then by definitions of $\phi$ and $A(x)$ we get that $\phi(A(x)\frac{x}{\|x\|})=0$. Hence by the mean value theorem and the fact that $\phi$ is $C^1$ due to

$$\phi\left(r\frac{x}{\|x\|}\right)\le M(A(x)-r)$$

for some constant $M$ and every $r$. As a result,

$$\int_0^{A(x)}\frac{dt}{\phi\left(t\frac{x}{\|x\|}\right)}$$

diverges. Hence we infer that $f([0,A(x))\frac{x}{\|x\|})=\mathbf{R}_+\frac{x}{\|x\|}$ and so $f(\Omega)=\mathbf{R}^n$.

To end the proof we need to show that $f$ has a $C^\infty$-inverse. But as a corollary from the inverse function theorem we get that it is sufficient to show that $df$ vanishes nowhere.

Suppose that $d_x f(h)=0$ for some $x\in\Omega$ and $h\neq 0$. From definition of $f$ we get that

$$d_x f(h)=\lambda(x)h+d_x \lambda(h)x.$$

Hence $h=\mu x$ for some $\mu\neq 0$ and from that $x\neq 0$. As a result $\lambda(x)+d_x \lambda(x)=0$. But we have that $\lambda(x)\ge1$ and function $g(t):=\lambda(tx)$ is increasing, so $g'(1)=d_x \lambda(x)\gt0$, which gives a contradiction.

Example

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

$$I(\Delta^n) = \left\{ (x^1, \cdots, x^n) \in \mathbb{R} | (\forall i : x^i \gt 0)\; and \; ( \sum_{i=1}^n x^i \lt 1) \right\} \,.$$

Then define the map $f : I(\Delta^n) \to \mathbb{R}^n$ by

$$(x^1, \ldots, x^n) \mapsto (\log(\frac{x^1}{1 - x^1 - \ldots -x^n}), \ldots, \log(\frac{x^n}{1 - x^1 - \ldots - x^n})) \,.$$

(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

$$\array{ \mathbb{R}^n & \stackrel{\exp^n}{\to} & \mathbb{R}_+^n & \to & I(\Delta^n) \\ & & \vec{x} & \mapsto & \vec{x}/(1 + {\|\vec{x}\|}_1) }$$

which we simply invert to get the map $f$ above.

Good covers by balls

One central application of balls is as building blocks for coverings. See good open cover for some statements.

Related concepts

• unit ball

References

Geometric

• V. Ozols, Largest normal neighbourhoods , Proceedings of the American Mathematical Society Vol. 61, No. 1 (Nov., 1976), pp. 99-101 (jstor)

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

• 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.

Star-shaped regions diffeomorphic to open ball

The proof that open star-shaped regions are diffeomorphic to a ball appears as

• Dirk Ferus, theorem 237 in: Analysis III (2007) (pdf, pdf)

It is a lengthy proof, due to Stefan Born.

A simpler version of the proof appears in

• Stéphane Gonnord, Nicolas Tosel, page 60 of: Calcul Différentiel, ellipses (1998) (English translation: MO:a/212595, pdf)

These proofs had remained obscure (see also this Remark at good open cover):

For instance in a remark below lemma 10.5.5 of

• Lawrence Conlon, Differentiable manifolds, Birkhäuser 2001/2008 (doi:10.1007/978-0-8176-4767-4)

it says:

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

And in

• Jeffrey Lee, Manifolds and differential geometry, Graduate Studies in Mathematics 107 (2009) (ISBN: 978-0-8218-4815-9, doi:10.1090/gsm/107)

one finds the statement:

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]).

Here “Grom” refers to

• Mikhail Gromov, Convex sets and Kähler manifolds, in F. Tricerri (ed.) Advances in differential geometry and topology, World Scientific (1990) 1-38 [pdf]

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.

A series of exercises covering the statement appears in

• Victor Guillemin, Peter Haine, Exercises 5.3.i–5.3.iv in: Differential Forms, World Scientific (2019) [doi:10.1142/11058]

See also the discussion at:

• Math Overflow, explicit diffeomorphim between open simplex and open ball [q/41853]

Combinatorial

• Andre Joyal, Disks, duality and Theta-categories (pdf)

• Mihaly Makkai, Marek Zawadowski, Duality for Simple $\omega$-Categories and Disks (TAC)