24 I. Convex Sets at Large

PROBLEMS.

1. Let S ⊂

Rd

be a compact convex set. Prove that there is a point u ∈

Rd

such that (−1/d)S + u ⊂ S.

Hint: For every point x ∈ S consider the set Ax = u : (−1/d)x + u ∈ S .

Use Helly’s Theorem.

2∗

(“Ham Sandwich Theorem”). Let μ1,... , μd be a set of Borel probability

measures on

Rd.

Prove that there exists a hyperplane H ⊂

Rd

such that μi(H+) ≥

1/2 and μi(H−) ≥ 1/2 for all i = 1, . . . , d.

3∗ (Center Transversal Theorem). Let μ1,... , μk, k ≤ d, be Borel probability

measures on Rd. Prove that there exists a (k − 1)-dimensional aﬃne subspace)(

L ⊂ Rd such that for every closed halfspace H containing L we have μi H ≥

1/(d − k + 2).

Remark: For Problems 2 and 3, see [Z97]

ˇ

and references therein.

4◦.

Prove that the theorem of Problem 3 implies both the result of Problem 2

above and the proposition of this section.

A couple of geometric problems.

5∗ (Krasnoselsky’s Theorem). Let X ⊂ Rd be a set and let a, b ∈ X be points.

We say that b is visible from a if [a, b] ⊂ X. Suppose that X ⊂ Rd is an infinite

compact set such that for any d+1 points of X there is a point from which all d+1

are visible. Prove that there is a point from which all points of X are visible.

6 (Jung’s Theorem). For a compact set X ⊂ Rd, let us call maxy,z∈X y − z

the diameter of X. Prove that any compact set of diameter 2 is contained in a ball

of radius 2d/(d + 1).

For Problems 5 and 6, see [DG63].

6. An Application to Approximation

We proceed to apply Helly’s Theorem to an important problem of constructing the

best approximation of a given function by a function from the required class. We

will go back to this problem again in Section IV.13.

(6.1) Uniform approximations.

Let us fix some real-valued functions fi : T −→ R, i = i, . . . , m, on some set

T . Given a function g : T −→ R and a number ≥ 0, we want to construct a linear

combination

fx(τ) =

m

i=1

ξifi(τ), x = (ξ1, . . . , ξm)

such that

|g(τ) − fx(τ)| ≤ for all τ ∈ T.

This is the problem of the uniform or Chebyshev approximation. Helly’s Theorem

implies that a uniform approximation exists if it exists on every reasonably small

subset of T .