Todd Trimble
Balls

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.

Shouldn’t the claim be obvious?

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

  • It’s surjective: DD contains a ball B=B ε(0)B = B_{\varepsilon}(0) in its interior, and for each nonzero 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.

  • 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.

I guess I’d now like to add a little more to this little article, since it started out being about compact convex sets.

Let DD be a compact convex set of n\mathbb{R}^n. The essential interior of DD is the set of points that are interior points of DD relative to its affine span. The essential boundary of DD is the set of points that do not belong to the essential interior of DD.

Lemma

If xx belongs to the essential boundary of DD, then there is some functional α: n\alpha: \mathbb{R}^n such that α(x)=1\alpha(x) = 1 and α(y)<1\alpha(y) \lt 1 for every essential interior point yy.

In this case, α 1(1)D\alpha^{-1}(1) \cap D is compact, convex, and has dimension strictly less than the dimension of DD.

Theorem

(Minkowski-Carathéodory) A compact convex DD subset of n\mathbb{R}^n is the convex hull of its extreme points. Indeed, every point xDx \in D is a convex combination of k+1k+1 of its extreme points if the affine span of DD is kk-dimensional.

This of course is a baby version of the Krein-Milman theorem.

Proof

By induction on kk. The case for k=0k=0 is trivial.

(to be continued)

References

Revised on September 21, 2013 at 04:22:11 by Todd Trimble