February 19, 2019 at 2:30 PM in Yost 306

Speaker: Florian Besau, Vienna Technical University

Abstract

For an origin-symmetric convex body K the boundary of the centroid body $\Gamma K$ is defined by the centroids of the intersection of K by half-spaces through the center of symmetry. This defines a compact convex subset $\Gamma K$ of K and $\Gamma$ operates affine-equivariant on convex bodies.

In $\R^3$ this notion was first considered by Blaschke, who conjectured that the volume ratio between $\Gamma K$ and K is minimized for ellipsoids.

This was later confirmed by Petty in all dimensions. Petty established a connection between the volume of the centroid body and the expect volume of random simplices. Hence, he was able to reinterpret Busemann’s random simplex inequality and established what is now known as the Busemann–Petty centroid inequality:

\frac{\vol_n(\Gamma K)}{\vol_n(K)} \geq \frac{\vol_n(\Gamma B)}{\vol_n(B)},

for all origin-symmetric convex bodies K and B denotes the Euclidean unit ball. Using the classical Blaschke–Santalo inequality one obtains the polar Busemann-Petty centroid inequality.

With the extension of the Brunn-Minkowski theory to the L_p- and Orlicz Brunn-Minkowski theory came also new notions of L_p- and Orlicz centroid bodies. These extensions lead to strong new L_p- and Orlicz Busemann-Petty centroid inequalities and numerous connections with asymptotic geometric analysis, geometric tomography, integral geometry and information theory have been uncovered.

In this talk I will present our definition for centroid bodies of centrally-symmetric (spherical) convex bodies of the Euclidean unit sphere. We show that two natural definitions, one geometric and the other probabilistic, lead to the same notion. We establish various properties of these spherical centroid bodies and obtain an inequality similar in nature to the polar Busemann-Petty centroid inequality in the spherical setting. At this point, a spherical version of the Busemann-Petty centroid inequality remains an important open conjecture.

This is based on joint work with Thomas Hack, Peter Pivovarov and Franz E. Schuster.