Georgia Institute of Technology College of Sciences

Let $K$ be a n dimensional convex body with of volume $1$. and barycenter of $K$ is the origin.  It is known that $|K \cap -K|>2^{-n}$.  Via thin shell estimate by Lee-Vempala (earlier versions were done by Guedon-Milman, Fleury, Klartag), we improve the bound by a sub-exponential factor.  Furthermore, we can improve  the Hadwiger’s Conjecture in the non-symmetric case by a sub-exponential factor.  This is a joint work with Boaz A. Slomka, Tomasz Tkocz, and Beatrice-Helen Vritsiou.