Millican Colloquium: Empirical forms of isoperimetric inequalities in convex geometry--Peter Pivovarov (University of Missouri) | Department of Mathematics

Millican Colloquium: Empirical forms of isoperimetric inequalities in convex geometry--Peter Pivovarov (University of Missouri)

Event Information
Event Location: 
GAB 461 (Refreshments at 3:30 in 472)
Event Date: 
Monday, October 17, 2016 - 4:00pm

In convex geometry there is a wealth of extremal inequalities relating fundamental metric quantities such as volume, surface area and mean width. These often arise by maximizing or minimizing a functional $\phi$ over all convex bodies of a given volume; for example, if $\phi(K)$ is the surface area of a convex body $K$, then minimization leads to the classical isoperimetric inequality with the Euclidean ball as the extremal case.

I will review several such inequalities including the isoperimetric inequality and the closely related Brunn-Minkowski inequality on the Minkowski (vector) sum of convex sets and their volume. I will then
discuss how these can be thought of as ``global'' inequalities that arise through a ``local'' random approximation procedure. This leads to randomized versions for which a stronger stochastic dominance
holds. Moreover, they recover the original inequalities via laws of large numbers. The approach involves a merger of techniques from stochastic geometry and analysis. I will explain how stochastic
dominance arises and its usefulness. The talk will be expository, with no special background assumed.