A solution to Bezdek's conjecture

For a given $\lambda >0$, a convex body in $\mathbb R^n$ is $\lambda$-convex if it is the intersection of (finitely or infinitely many) balls of radius $1/\lambda$. In this note, we show that among all $\lambda$-convex bodies in $\mathbb R^n$, $n \geqslant 2$, with a given inradius, the $\lambda$-convex lens (i.e., the intersection of two balls of radius $1/\lambda$) has the largest mean width. This gives an affirmative answer to the conjecture of K. Bezdek. Under an additional symmetry assumption on $\lambda$-convex bodies, we resolve the analogous inradius conjecture of Bezdek for arbitrary intrinsic volumes. We also establish an answer to the corresponding conjecture of K. Bezdek about the circumradius. In particular, we prove that the $\lambda$-convex spindle (i.e., the intersection of all balls of radius $1/\lambda$ containing a given pair of points) is the unique minimizer of the mean width among all $\lambda$-convex bodies with a fixed circumradius.