  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
$\mathbb{R}^n$.

Metric Geometry;Differential Geometry