A complete solution to the Grünbaum--Loewner centroid problems

Abstract

In the 1960s, Grünbaum and Loewner raised problems concerning the minimum possible number of hyperplane sections of a convex body in $R^n$ whose centroids coincide with the centroid of the body itself. By employing the hemispherical transform and Morse theory, we provide a complete solution to these problems.

18 pages, 1 figure. This version has three authors. The results in this paper are obtained independently and simultaneously by Shouda Wang (at Princeton University) and by Ge Xiong and Kaiwen Yang (at Tongji University)