Reverse isoperimetric problems under curvature constraints

Geometry Topology Seminar
Friday, March 17, 2023 - 11:00am for 1 hour (actually 50 minutes)
Skiles 006
Kateryna Tatarko – University of Waterloo – ktatarko@uwaterloo.ca
Galyna Livshyts

Please Note: Note the unusual time!

In this talk we explore a class of $\lambda$-convex bodies, i.e., convex bodies with curvature at each point of their boundary bounded below by some $\lambda >0$. For such bodies, we solve two reverse isoperimetric problems.

In $\mathbb{R}^3$, we show that the intersection of two balls of radius $1/\lambda$ (a $\lambda$-convex lens) is the unique volume minimizer among all $\lambda$-convex bodies of given surface area.  We also show a reverse inradius inequality in arbitrary dimension which says that the $\lambda$-convex lens has the smallest inscribed ball among all $\lambda$-convex bodies of given surface area.

This is a joint work with Kostiantyn Drach.