Tuesday February 13, 2018 3:00pm – 4:00pm, Yost Hall, Room 306

Speaker: Olaf Mordhorst, Vienna Technical University

Abstract: In this talk we present a duality relation between floating and illumination bodies. The convex floating body was introduced independently by B\'{a}r\'{a}ny/Larman and Sch\”utt/Werner for the study of random polytopes and for the extension of affine surface area to the class of all convex bodies. Later, E. Werner introduced the illumination body which has similar properties as the floating body. The definition of these two notions suggests that the polar of a floating body should not be too far away from the illumination body of the polar of a convex body although equality cannot be achieved in general. We consider this question for the class of centrally symmetric convex bodies and we provide asymptotically sharp estimates for the distance of the polar of the floating body to the illumination body of the polar. This distance has a connection to other important notions in convex geometry, namely, the cone measure of a convex body. Furthermore, our estimates show that ellipsoids are the only example where equality holds.

The talk is based on joint work with E. Werner.

Partners: Dr. Elisabeth Werner, Dr. Mark Meckes, Dr. Elizabeth Meckes, Dr. Wojbor Woyczynski