Harmonic Analysis and Convex Geometry, Fall 2013, August 30.

Instructor: Dmitry Ryabogin

Assignment 1.

1. Problem 1. Prove the Radon Theorem: Each set of affinely dependent point (in particular, each set of at least n + 2 points) in ℝ^{n} can be expressed as the union of two disjoint sets whose convex hulls have a common point.

2. Problem 2. For A ⊂ ℝ^{n} the set of all convex combinations of any finitely many elements of A is called the convex hull of A and is denoted by convA.

a) Prove the Caratheodory Theorem: If A ⊂ ℝ^{n} and x ∈ convA, then x is a convex combination of affinely independent points of A. In particular, x is a convex combination of n + 1 or fewer points in A.

Hint: The point x ∈ convA has a representation

x = ∑_{j=1}^{k} λ_{j}x_{j}, x_{j} ∈ A, λ_{j} > 0, ∑_{j=1}^{k} λ_{j} = 1

with some k ∈ ℕ. Assume that k is minimal. Using the definition of an affine dependence obtain an affine representation of x with non-negative coefficients at least one of which is zero. Get a contradiction with a minimality of k.

b) Prove that in ℝ^{n}, the convex hull of a compact set is compact.

Hint: Let A ⊂ ℝ^{n} be a compact set and let S be the set of points

α = (α_{0}, α_{1}, …, α_{n}) ∈ ℝ^{n+1} : ∑_{j=0}^{n} α_{j} = 1, α_{j} ≥ 0.

For each point

(α, x) := ((α_{0}, α_{1}, …, α_{n}), (x_{0}, x_{1}, …, x_{n})) ∈ S × A^{n+1}

let f(α, x) = ∑_{j=0}^{n} α_{j}x_{j}. Prove that f(S×A^{n+1}) is compact. Conclude that f(S×A^{n+1}) =

convA.

3. Problem 3. Let M be a finite family of convex sets in ℝ^{n} and let K ⊂ ℝ^{n} be convex. If any n + 1 elements of M are intersected by some translate of K, then all elements of M are intersected by a translate of K.

4. Problem 4. Let A ⊂ ℝ^{n} be a set with diam A ≤ 2. Prove that A lies in a Euclidean ball of radius \sqrt{\frac{2n}{n+1}}. If A does not lie in any smaller ball, then the closure of A contains the vertices of a regular n-simplex of edge-length 2.

Hint: By the previous problem we can assume that the cardinality of A, card A ≤ n+1. In this case, let y be the center of the smallest Euclidean ball containing A and let r = r(A) be its radius. Denote

{z_{0}, …, z_{m}} = {x ∈ A : |y − x| = r}, m ≤ n.

Prove that y ∈ conv{z_{0}, …, z_{m}}. We can assume that y = 0. Hence,

0 = ∑_{j=0}^{m} λ_{j}z_{j},  ∑_{j=0}^{m} λ_{j} = 1,  λ_{j} ≥ 0.

Since 4 ≥ |z_{k} − z_{j}|^{2} = 2r^{2} − 2z_{k} · z_{j}, show that

1 − λ_{k} ≥ ∑_{j=0}^{m} λ_{j} \frac{|z_{j} − z_{k}|^{2}}{4} ≥ \frac{r^{2}}{2} ,

and conclude that r^{2} ≤ \frac{2m}{m+1}. What happens if we have 1 − λ_{k} = \frac{r^{2}}{2} ?