24 I. Convex Sets at Large
1. Let S
be a compact convex set. Prove that there is a point u
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.
(“Ham Sandwich Theorem”). Let μ1,... , μd be a set of Borel probability
measures on
Prove that there exists a hyperplane H
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]
and references therein.
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
fx(τ) =
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 .
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