Convexity 2
Homework 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.
- 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.
- 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.
- Prove that a convergent sequence of balls must converge either to a ball or to a point.
- 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.
- 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.