Harmonic Analysis and Convex Geometry, Fall 2013, September 27.

Instructor: Dmitry Ryabogin

Assignment 5.

1. Problem 1.

a) Prove that any origin-symmetric polytope in ℝ^{d} is a projection of a linear image of the octahedron in ℝ^{f}. What is f?

c) Prove that for a d-dimensional simplicial polytope P we have ∑_{j=0}^{d} f_{j}(P) ≤ 2^{d}f_{d−1}(P).

2. Problem 2. Prove that there are only five regular polytopes in ℝ^{3} (a polytope is called regular if the amount of edges incident to each vertex is equal to the amount of faces incident to the vertex and all the vertices are equivalent).

3. Problem 3.

a) Prove that any ellipsoid in ℝ^{3} centered at the origin has a disc as one of its central sections (by the plane passing through the origin).

b) Prove that any d-dimensional ellipsoid in ℝ^{d} centered at the origin has a k-dimensional ball as one of its central sections (by the k-dimensional subspace). What is k?

4. Problem 4. Let C ⊂ ℝ^{d} be a cone starting at the origin, and let S^{d−1} be a unit sphere in ℝ^{d}, S^{d−1} = {x ∈ ℝ^{d} : |x| = 1}. Define a solid angle σ(C) of C to be a (d − 1)-volume of the intersection: σ(C) := vol_{d−1}(C ∩ S^{d−1}). Let v be a vertex of a convex polytope P in ℝ^{d}. Define the curvature ω_{v} of v to be a solid angle σ(C^{∗}) of the dual cone C^{∗} of v, i.e., C^{∗} = {x ∈ ℝ^{d} : x = λn_{v}, λ ≥ 0}, where n_{v} are the outer normals to the supporting hyperplanes of P at v.

Prove the Gauss-Bonnet Theorem: the sum of the curvatures of all the vertices of P is the (d − 1)-volume of S^{d−1}.

5. Problem 5*. Let P be a polytope in ℝ^{d} containing the origin in its interior, and let F be an r-dimensional face of P, 0 ≤ r ≤ d − 1. Denote by V (F) an (r + 1)-dimensional subspace containing F. The polytope P is called r-equatorial, if for every F, P ∩ V (F) is the union of r-dimensional faces of P (B_{1}^{3} is 1-equatorial, B_{∞}^{3} is not; any P as above is (d − 1)-equatorial, and any origin-symmetric one is 0-equatorial). Prove that P is r-equatorial if and only if the (d − r − 1)-dimensional faces of the (d − r)-dimensional faces of P^{∗} occur in parallel pairs.