A complete solution to the reverse isoperimetric problem for $λ$-convex bodies

Abstract

For a given $λ> 0$, a convex body in $\mathbb{R}^n$ is called $λ$-convex if it is the intersection of balls of radius $1/λ$. We show that among all $λ$-convex bodies in $\mathbb R^n$ of a given surface area, volume is minimized by the $λ$-convex lens, that is, the intersection of two balls of radius $1/λ$. This confirms a conjecture of Borisenko in all dimensions $n \geq 2$. Moreover, we extend this reverse isoperimetric inequality to all model spaces of constant curvature in arbitrary dimension and arbitrary $λ> 0$.