nLab
ball

Context

Topology

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

Contents

Definition

Geometric

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

D n:={x n| i(x i) 21} n 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 n\mathbb{R}^n.

Its interior is the open nn-ball

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

Its boundary is the (n1)(n-1)-sphere.

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

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

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

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 DD in n\mathbb{R}^n with nonempty interior is homeomorphic to D nD^n.

Proof

Without loss of generality we may suppose the origin is an interior point of DD. We claim that the map ϕ:vv/v\phi: v \mapsto v/{\|v\|} maps the boundary D\partial D homeomorphically onto S n1S^{n-1}. By convexity, DD is homeomorphic to the cone on D\partial D, and therefore to the cone on S n1S^{n-1} which is D nD^n.

The claim reduces to the following three steps.

  1. The restricted map ϕ:DS n1\phi: \partial D \to S^{n-1} is continuous.

  2. It’s surjective: DD contains a ball B=B ε(0)B = B_{\varepsilon}(0) in its interior, and for each xBx \in B, the positive ray through xx intersects DD in a bounded half-open line segment. For the extreme point vv on this line segment, ϕ(v)=ϕ(x)\phi(v) = \phi(x). Thus every unit vector uS n1u \in S^{n-1} is of the form ϕ(v)\phi(v) for some extreme point vDv \in D, and such extreme points lie in D\partial D.

  3. It’s injective: for this we need to show that if v,wDv, w \in \partial D are distinct points, then neither is a positive multiple of the other. Supposing otherwise, we have w=tvw = t v for t>1t \gt 1, say. Let BB be a ball inside DD containing 00; then the convex hull of {w}B\{w\} \cup B is contained in DD and contains vv as an interior point, contradiction.

So the unit vector map, being a continuous bijection DS n1\partial D \to S^{n-1} between compact Hausdorff spaces, is a homeomorphism.

Corollary

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

Proof

DD has nonempty interior relative to its affine span which is some kk-plane, and then DD is homeomorphic to D kD^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 nn-ball is homeomorphic and even diffeomorphic to the Cartesian space n\mathbb{R}^n

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

For instance, the smooth map

xx1+|x| 2: n𝔹 n x\mapsto \frac{x}{\sqrt{1+|x|^2}} : \mathbb{R}^n \to \mathbb{B}^n

has smooth inverse

yy1|y| 2:𝔹 n n. y\mapsto \frac{y}{\sqrt{1-|y|^2}} : \mathbb{B}^n \to \mathbb{R}^n.

This probe from n{\mathbb{R}}^n witnesses the property that the open nn-ball is a (smooth) manifold. Hence, each (smooth) nn-dimensional manifold is locally isomorphic to both n{\mathbb{R}}^n and 𝔹 n\mathbb{B}^n.

From general existence results about smooth structures on Cartesian spaces we have that

Theorem

In dimension dd \in \mathbb{N} for d4d \neq 4 we have:

every open subset of d\mathbb{R}^d which is homeomorphic to 𝔹 d\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 4\mathbb{R}^4. See De Michelis-Freedman.

Theorem

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

Remark

This is a folk theorem. But explicit proofs in the literature are very hard to find. See the discussion at References. An explicit proof has been written out by Stefan Born, and this appears as the proof of theorem 237 in (Ferus).

Proof

Suppose TT is a star-shaped open subset of n{\mathbb {R}}^n centered at the origin. Theorem 2.29 in Lee proves that there is a function ff on n{\mathbb{R}}^n such that f>0f\gt 0 on TT and ff vanishes on the complement of TT. By applying bump functions we can assume that f1f\le 1 everywhere and f=1f=1 in an open ϵ\epsilon-neighborhood of the origin; by rescaling the ambient space we can assume ϵ=2\epsilon=2.

The smooth vector field V:xf(x)x/xV\colon x\mapsto f(x)\cdot x/{\|x\|} is defined on the complement of the origin in TT. Multiply VV by a smooth bump function 0b10\le b\le 1 such that b=1b=1 for x>1/2{\|x\|} \gt 1/2 and b=0b=0 in a neighborhood of 0. The new vector field VV extends smoothly to the origin and defines a smooth global flow F:×TTF\colon \mathbb{R} \times T\to T. (The parameter of the flow is all of \mathbb{R} and not just some interval (,A)(-\infty,A) because the norm of VV is bounded by 1.) Observe that for 1/2<x<21/2\lt {\|x\|} \lt 2 the vector field VV equals xx/xx\mapsto x/{\|x\|}. Also, all flow lines of VV are radial rays.

Now define the flow map p: >1/2 nT >1/2p\colon{\mathbb{R}}^n_{\gt 1/2}\to T_{\gt 1/2} as xF(x1,xx)x\mapsto F({\|x\|}-1, \frac{x}{{\|x\|}}) for x>1/2{\|x\|} \gt 1/2. (The subscript >1/2\gt 1/2 removes the closed ball of radius 1/21/2.) The flow map is the composition of two diffeomorphisms,

>1/2 n(1/2,)×S n1T >1/2,{\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)TL: (A,B) \to T, where AA and BB can be finite or infinite. If BB is finite and the limit of L(t)L(t) as tBt \to B exists, then the vector field VV vanishes at BB. In our case VV can only vanish at the boundary of TT, which is precisely what we want for surjectivity.)

Finally, define the desired diffeomorphism d: nTd\colon{\mathbb{R}}^n\to T as the gluing of the identity map for x<2{\|x\|} \lt 2 and as pp for x>1/2{\|x\|}\gt 1/2. The map gg is smooth because for 1/2<x<21/2\lt {\|x\|} \lt 2 both definitions give the same value.

Example

Let I(Δ n) nI(\Delta^n) \subset \mathbb{R}^n be the interior of the standard nn-simplex. Then there is a diffeomorphism to 𝔹 n\mathbb{B}^n defined as follows:

Parameterize the nn-simplex as

I(Δ n)={(x 1,,x n)|(i:x i>0)and( i=1 nx i<1)}. 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(Δ n) nf : I(\Delta^n) \to \mathbb{R}^n by

(x 1,,x n)(log(x 11x 1x n),,log(x n1x 1x n)). (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(Δ n)I(\Delta^n) is the positive orthant of an open nn-ball in l 1l^1 norm, so that in the opposite direction we have a chain of invertible maps

n exp n + n I(Δ n) x x/(1+x 1)\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 ff above.

Good covers by balls

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

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

  • De Michelis, Stefano; Freedman, Michael H. (1992) “Uncountably many exotic 4\mathbb{R}^4‘s in standard 4-space”, J. Diff. Geom. 35, pp. 219-254.

The proof that open star-shaped regions are diffeomorphic to a ball appears as theorem 237 in

It is a lengthy proof, due to Stefan Born.

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

  • Stéphane Gonnord, Nicolas Tosel, Calcul Différentiel , ellipses (1998)

Apparently this proof is little known. For instance in a remark below lemma 10.5.5 of

  • Lawrence Conlon, Differentiable manifolds, Birkhäuser (last edition 2008)

it says:

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

And in

  • Jeffrey Lee, Manifolds and differential geometry (2009)

one finds the statement:

Actually, the assertion that an open geodesically convex set in a Riemannian manifold is diffeomorphic to n\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

  • M. Gromov, Convex sets and Kähler manifolds , Advances in differential geometry and topology. F. Tricerri ed., World Sci., Singapore, (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 1C^1 class, not C C^\infty, so that this is not a proof, either.

For a discussion of diffeomorphisms between geodesically convex regions and open balls see good open cover.

See also the Math Overflow discussion here.

Combinatorial

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

Revised on August 3, 2015 09:23:32 by Todd Trimble (67.81.95.215)