Georgia Institute of Technology College of Sciences

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.