Convexity 1 - Homework 3

 

Undergraduate questions:

  1. Lay Question 3.2
  2. Lay Question 3.4
  3. Lay Question 3.7
  4. Lay Question 3.8
  5. Suppose that A1 is a subset of the hyperplane H1 and that A2 is a subset of the parallel hyperplane H2. Prove that A1+A2 is subset of a hyperplane parallel to both H1 and H2.
Additional graduate questions:
 
  1. In E4 let K be the convex hull of the usual basis vectors e1, e2, e3, e4. Prove that K lies in a hyperplane and give an equation of this hyperplane. What is the three dimensional volume of K?

  2. Suppose that A is a subset of a closed half space determined by the hyperplane H in En. Prove that the convex hull of the intersection of H and A is the intersection of H with the convex hull of A.