Convexity 2

Homework 1

  1. Let E be an ellipse with major and minor axes of lengths 2a and 2b. Let F be the convex body obtained by rotating E through ninety degrees about its center. Calculate the Hausdorff distance between E and F, giving full explanations.
  2. Let H1, H2, ... be a sequence of supporting hyperplanes to a convex body K contained in the Ball rB. Assume that the sequence of intersections of the Hi with rB converge. Prove that, if R > r, the sequence of intersections of the Hi with RB also converges. If the limit of this sequence is denoted by T(R), prove that T(R) is the intersection of a support hyperplane of K with RB.
  3. Assume that the convex bodies K1, K2, ... converge to K. Let xi be a point of Ki for each i=1, 2, ... . Show that the sequence of points x1, x2, ... contains a subsequence convergent to a point of K.
  4. Prove that a convergent sequence of balls must converge either to a ball or to a point.
  5. Assume that the convex bodies K1, K2, ... converge to K. If each Ki is contained in the convex body P and contains the convex body Q, prove that K is contained in P and contains Q.
  6. Assume that the convex bodies K1, K2, ... converge to K. Assume further that S is a closed set which does not intersect the interior of any Ki. Prove that S does not intersect the interior of K.