Harmonic Analysis and Convex Geometry, Fall 2013, September 20.

Instructor: Dmitry Ryabogin

Assignment 4.

1. Problem 1. Let M and L be two sets in ℝ^{n}. A set

M + L = {x ∈ ℝ^{n} : x = x_{1} + x_{2}, x_{1} ∈ M, x_{2} ∈ L}

is a sum of sets M and L and is denoted by M + L.

a) Prove that M + L is a polytope, provided M and L are.

b) Is it true that 2A = A + A for any set A? for any convex set A?

c) Is it true that A + A convex implies A is convex? d)* Let convex bodies M and L be such that M + L is a polytope. Is it true that M and L are polytopes? e)* Let A and B be two convex polytopes. What is the polar of A + B?

f) Let M = B_{3}^{1} + B_{3}^{∞}. How many vertices, edges, faces does M have?

2. Problem 2. Let M ⊂ ℝ^{d} and L ⊂ ℝ^{n}. A product of M and L is a set defined as

M × L = {(x_{1}, x_{2}) : x_{1} ∈ M, x_{2} ∈ L}.

a) Prove that a product of finitely many polytopes is a polytope. b)* Let M and L be two polytopes. What is the polar of M × L?

c) Let M and L be two polygons in ℝ^{2}. How many edges does M × L ∈ ℝ^{4} have?

3. Problem 3.

a) Let K = [−5, −4]. Find (K^{∗})^{∗}.

b) Prove that (AK)^{∗} = (A^{t})^{−1}K^{∗} for any linear transformation A ∈ GL(n) and a convex body K such that 0 ∈ int K.

c) Let K and L be two convex symmetric bodies. Prove that (L∩K)^{∗} = conv (K^{∗}, L^{∗}).

d) Prove that the intersection of finitely many polytopes is a polytope.

e) Prove that the affine image of a polytope is a polytope. Hint: α(M) = convα(vertM).

4. Problem 4*. Let K ∈ ℝ^{3} be a convex body such that all of its projections onto two-dimensional subspaces (planes passing through the origin) are symmetric. Prove that K is symmetric. Formulate and prove the analogous result in ℝ^{n}.