Harmonic Analysis and Convex Geometry, Fall 2013, October 11.

Instructor: Dmitry Ryabogin

Assignment 7.

1. Problem 1.

a) Let A be a convex set in ℝ^{d−1}, d ≥ 2, and let x = (x_{1}, …, x_{d−1}, x_{d}) ∈ ℝ^{d}, x_{d} ≠ 0. Prove that vol_{d}(conv(A, x)) = \frac{1}{d}|x_{d}|vol_{d−1}(A).

b) Compute the volume of the Euclidean unit ball vol_{d}(B_{2}^{d}).

c) Let ω_{d−1} stand for the (d − 1)-dimensional surface area measure of the Euclidean unit ball B_{2}^{d}, d ≥ 2. Prove that ω_{d−1} = dvol_{d}(B_{2}^{d}).

d) Prove that for all n ∈ S^{d−1}, ∫_{S^{d−1}} |n · v|dσ_{d−1}(v) = 2vol_{d−1}(B_{2}^{d−1}).

Hint: The left-hand side is precisely twice the (d − 1)-dimensional volume of the projection of the unit ball.

2. Problem 2.

a) Let P be a convex polytope in ℝ^{d} and let u^{⊥} be a (d−1)-dimensional subspace of ℝ^{d} orthogonal to a direction u ∈ S^{d−1}. Prove that 2vol_{d−1}(P |u^{⊥}) = ∑_{j=1}^{k} |u · n_{j}|vol_{d−1}(F_{j}), where F_{j}, j = 1, …, k, are the facets of P, and n_{j} are their normals.

b) Prove the Cauchy surface area formula: S(P) = \frac{1}{vol_{d−1}(B_{2}^{d−1})} ∫_{S^{d−1}} vol_{d−1}(P |u^{⊥})dσ(u), where S(P) is the sum of the (d − 1)-dimensional volumes of the facets of P. Hint: Use d) of the previous exercise.

3. Problem 3. A convex polytope Z in ℝ^{d} is called a zonotope if it is a Minkowski sum

of segments, Z = ∑_{j=1}^{k}[a_{j}, b_{j}], a_{j}, b_{j} ∈ ℝ^{d}.

a) Prove that a zonotope is always centrally-symmetric.

b) Prove that a translation of a linear image of a zonotope is a projection of the k-dimensional cube.

c) Let F (Z, u) be a face of a zonotope Z orthogonal to a direction u ∈ S^{d−1} (F is the intersection of Z and the supporting hyperplane orthogonal to u). Prove that F (Z, u) = ∑_{j=1}^{k} F ([a_{j}, b_{j}], u). Conclude that any face of a zonotope is itself a zonotope.

d)* Prove that a polytope in ℝ^{d} is a zonotope iff all its two-dimensional faces are symmetric. e)* Prove that a polar of a zonotope is not a zonotope.

4. Problem 4*. Let P be a convex polytope in ℝ^{3} with 7 facets. The area of each facet is 1. Is it possible to estimate the volume of P from below by some fixed positive constant?