Kostiantyn Drach, Universitat de Barcelona
Reverse inradius inequalities for ball-bodies
A ball-body, also called a $\lambda$-convex body, is an intersection of congruent Euclidean balls of radius $1/\lambda$ in $\mathbb{R}^n$, $n \geq 2$. Such bodies arise naturally in optimization problems in combinatorial and convex geometry, in particular when the number of generating balls is finite. In recent years, ball-bodies have also played a central role in an active research program on reverse isoperimetric-type problems under curvature constraints. The general objective of this program is to understand how prescribed curvature bounds restrict the extremal behavior of geometric functionals (e.g., volume, surface area, or mean width), and to identify sharp inequalities between them that reverse the existing classical isoperimetric-type inequalities. In this talk, we focus on the inradius minimization problem for $\lambda$-convex bodies with prescribed surface area or prescribed mean width. Here, the inradius of a convex body $K$ is the radius of the largest ball contained in $K$. In this setting, we establish sharp lower bounds for the inradius and show that equality is attained only by lenses, that is, intersections of two balls of radius $1/\lambda$. This solves a conjecture of Karoly Bezdek. We will outline the main ideas of the proof and pose several open problems. This is joint work with Kateryna Tatarko.
MC 5417