Reverse isoperimetric problems under curvature constraints

Abstract: In this paper we solve several reverse isoperimetric problems in the class of $\lambda$-convex bodies, i.e., convex bodies whose curvature at each point of their boundary is bounded below by some $\lambda > 0$. We give an affirmative answer in $\mathbb{R}^3$ to a conjecture due to Borisenko which states that the $\lambda$-convex lens, i.e., the intersection of two balls of radius $1/\lambda$, is the unique minimizer of volume among all $\lambda$-convex bodies of given surface area. Also, we prove a reverse inradius inequality: in model spaces of constant curvature and arbitrary dimension, we show that the $\lambda$-convex lens (properly defined in non-zero curvature spaces) has the smallest inscribed ball among all $\lambda$-convex bodies of given surface area. This solves a conjecture due to Bezdek on minimal inradius of isoperimetric ball-polyhedra in $\mathbb{R}^n$.