The ACO Seminar (2025–2026)

Given a body (compact, connected set with non-empty interior) K in n-dimensional Euclidean space, a natural question is: if one partitions the body into two pieces along its barycenter, how small can each piece be? By “partition along its barycenter”, we mean intersecting K with a half-space whose boundary is a hyperplane containing said barycenter. An easy observation is that, if K is symmetric about a point, then each piece will have (1/2) the total volume.

Grünbaum showed that, if K is convex, then the volume of each piece is at least (n/(n+1))^n times the total volume of K. Furthermore, this constant is sharp: there is equality if and only if K is a cone, which means there exists a (n − 1)-dimensional convex body L and a vector b, such that K has face L and vertex b (we say K is the convex hull of b and L). Notice the number (n/(n+1))^n is greater than (1/e), and in fact approaches it as the dimension goes to infinity. That is, the general situation, using constant (1/e), is not much worse than the symmetric case.

In this work, which is joint with M. Fradelizi, J. Liu, F. Marin Sola, and S. Tang, we are interested in generalizing Grünbaum’s inequality to other measures. Our main results are a sharp inequality for the Gaussian measure and a sharp inequality for s-concave probability measures. The characterization of the equality case is of particular interest.