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 of the equality case is of particular interest.
