Harmonic Analysis and Convex Geometry, Fall 2013, September 6.

Instructor: Dmitry Ryabogin

Assignment 2.

1. Problem 1.

a) Let W be a convex set. A point x ∈ W is called exterior if it does not lie on any segment with ends in W. Prove that the vertices of the polytope are the only exterior points of it.

b*) Prove that every non-empty compact convex set has an extreme point.

Hint: Use induction on dimension.

2. Problem 2. Let M be a polytope and let W ⊂ V = vert M. Then conv W is a face of M if and only if aff W ∩ conv (V \ W) = ∅ (here aff W stands for the affine hull of W).

3. Problem 3.

a) Prove that every ridge of a polytope is an intersection of two facets.

Hint: Let F_{k} be a facet of an H-polytope M. Then

F_{k} = H_{k} ∩ ( ⋂_{j≠k} H_{j}^{+}) = ⋂_{j≠k}(H_{k} ∩ H_{j}^{+}).

Hence,

R = F_{k} ∩ relbd (H_{k} ∩ H_{j}^{+}) = F_{k} ∩ H_{k} ∩ H_{j} = K ∩ H_{k} ∩ H_{j} = F_{k} ∩ F_{j}.

b) Let F = F^{j}, be a proper j-face of a d-polytope M, dim F^{j} = j and let j ≤ k ≤ d−1. Prove that F is an intersection of at least k − j + 1 k-faces containing F.

Hint: Prove at first that for any F there is a facet containing it: take x ∈ rel int F^{j} and observe that x must belong to a supporting hyperplane H generating some facet F^{d−1}. Prove that F^{j} ⊂ H and conclude that F^{j} ⊂ F^{d−1}. Then proceed by induction. Use a) to prove the second part of the theorem: each (k − 1)-dimensional face of F^{k+1} is an intersection of two k-dimensional faces.

c) Let F be a face of a d-polytope M ⊂ ℝ^{d}, dim F = j. Prove that there exists a (d − j − 1)-dimensional face F′ of M such that dim conv (F ∪ F′) = d.

Hint: In the case j ≤ d − 2 use induction, the fact that each face of a polytope is contained in some facet and a).

4. Problem 4*. Let K be a convex body in ℝ^{3} such that its orthogonal projections onto all planes (passing through the origin) are polygons. Prove that K is a polytope.