### Mathematics Colloquium

** Dmitri Ryabogin
**

Kent State University

**Title:** A solution to a local version of the fifth Busemann-Petty Problem.

**Date:** Friday, September 27, 2019

**Place and Time:** Room 101, Love Building, 3:35-4:25 pm

**Refreshments:** Room 204, Love Building, 3:00 pm

**Abstract.**
In 1956, Busemann and Petty posed a series of questions about
symmetric convex bodies, of which only the first one has been solved.
Their fifth problem asks the following.
Let K be an origin symmetric convex body in the n-dimensional Euclidean
space and let H_x be a hyperplane passing through the origin orthogonal to
a unit direction x. Consider a hyperplane G parallel to H_x and supporting
to K and let
C(K,x)=vol(K\cap H_x)dist (0, G).
If there exists a constant C such that for all directions x we have
C(K,x)=C, does it follow that K is an ellipsoid?
We give an affirmative answer to this problem for bodies sufficiently
close to the Euclidean ball in the Banach Mazur distance. This is a joint
work with Maria Alfonseca, Fedor Nazarov and Vlad Yaskin.