Convexity 2

Homework 2

1. Assume x is a point not in the convex body K. Prove that the hyperplane containing p(K,x) and normal to the vector x-p(K,x) supports K.
2. Assume r is any non-negative number and K is a convex body. Prove that K+rB is the set of points x for which d(x,p(K,x)) is at most r.
3. Let P be a polytope in n-dimensional space and let F be a facet of P. If r is non-negative, find an expression for the volume of the set of points x in P+rB for which p(P,x) is in the relative interior of F.
4. Lay question 19.3.
5. Lay question 19.4 parts c and d.