Harmonic Analysis and Convex Geometry, Fall 2013, October 25.

Instructor: Dmitry Ryabogin

Assignment 8.

1. Problem 1.

A boundary point x of a convex body K ⊂ ℝ^{n} is said to be singular if more than one supporting hyperplane of the body goes through it. If only one supporting hyperplane H_{x} passes through x, the boundary point is called regular.

a) Prove that if a convex body has only regular boundary points, then its boundary can be mapped continuously onto the surface of the n-dimensional unit ball by means of equally directed supporting hyperplanes. If, in addition, K ∩ H_{x} = {x} for every boundary point x ∈ K, prove that this map is one to one.

Let K be a convex body in ℝ^{n}, n ≥ 2, and let x be one of its boundary points. Consider the collection of rays emanating from x which contain a point of the body different from x, and adjoint the accumulation points of the point set that arises. The resulting set is a convex cone with vertex x, which is called the projection cone of the body K at the point x.

b) Prove that every supporting hyperplane passing through x is a supporting hyperplane of the cone, and conversely.

c) Conclude that the projection cone can be obtained as the intersection of all the half-spaces containing the body K whose boundary hyperplanes pass through x.

d) What is the projection cone at a regular point ?

2. Problem 2.

Let K be an origin-symmetric convex body in ℝ^{n}, n ≥ 2, let F (x) = ∥x∥_{K}, let x be a boundary point of K, and let y be a variable point.

a) Prove that the points y that satisfy F′(x; y − x) ≤ 0 are exactly those that form the projection cone of the body at the point x.

b) Prove that if x is regular, then F′(x; y − x) must be a linear function of y − x.

c) Conclude that if a convex body has only regular boundary points, then its norm has continuous first partial derivatives.

3. Problem 3.

A normal cone (or a polar cone) of a convex body K ⊂ ℝ^{n}, n ≥ 2, at the boundary point x is obtained if one erects at x the normals to all the supporting hyperplanes of K at x on the side turned away from the body. If the normal cone at x is (n − p)- dimensional, 0 ≤ p ≤ n, then x is called a p-face point. a) Prove that the projection cone at a p-face point x contains a p-dimensional flat (affine subspace) going through x and no flat of higher dimension.

b) Prove that if F′(x, y) has n linearly independent linearity directions, then it is linear.

c) Let F be as in Problem 2 and let x be a p-face point of K. Prove that F′(x; y) has, as a function of y, exactly p + 1 linearly independent linearity directions, and conversely.

4. Problem 4*. Let K be an origin-symmetric convex body in ℝ^{n}, n ≥ 3. Assume also that all sections (by the hyperplanes passing through the origin) of K are absolutely symmetric. Is it true that K is a Euclidean ball?