Harmonic Analysis and Convex Geometry, Fall 2013, September 13.

Instructor: Dmitry Ryabogin

Assignment 3.

1. Problem 1. Given a polytope M in ℝ^{n}, let f_{j}(M) be the numbers of its j-dimensional faces, 0 ≤ j ≤ n − 1. Denote by S a regular simplex, by B_{n}^{∞} := conv({v_{j}}_{j=1}^{2^{n}}) a standard cube (here v_{j} are all possible vectors having ±1 as its coordinates), and by B_{n}^{1} := conv(±e_{1}, …, ±e_{n}) a cross-polytope. Compute f_{j}(S), f_{j}(B_{n}^{∞}), and f_{j}(B_{n}^{1}), 0 ≤ j ≤ n − 1.

2. Problem 2. Prove that any origin-symmetric convex polytope M ⊂ ℝ^{n} with 2f facets is a section of a d-dimensional cube for some large d. What is d?

3. Problem 3. The (n−1)-dimensional permutahedron is the convex hull of the n! vectors in ℝ^{n} arising by permuting the coordinates of (1, 2, …, n).

a) Verify that it really has n! vertices corresponding to the permutations of {1, 2, …, n}.

b) Describe all faces of the permutahedron combinatorially (what sets of permutations are vertex sets of faces?).

c) Determine the dimensions of the faces found in b). In particular, show that the facets correspond to ordered partitions (A, B) of {1, 2, …, n}, A, B ≠ ∅, and count them.

4. Problem 4*. Let S be a (not necessarily regular) simplex in ℝ^{n}. For every edge e of S consider the n-dimensional ball B_{e} having e as its diameter. Show that each x ∈ S is covered by at least n balls B_{e}.

5. Problem 5*. Let C_{1} be a unit cube in ℝ^{3}, and let C_{a}, C_{b} be cubes with edges of length a and b. Prove that a + b ≤ 1, provided C_{a} ∪ C_{b} ⊂ C_{1}, int(C_{a}) ∩ int(C_{b}) = ∅. Can you solve a similar problem in ℝ^{n}?