On the isodiametric and isominwidth inequalities for planar bisections

Abstract

For a given planar convex body $K$, a bisection of $K$ is a decomposition of $K$ into two closed sets $A,B$ so that $A\cap B$ is an injective continuous curve connecting exactly two boundary points of $K$. Consider a bisection of $K$ minimizing, over all bisections, the maximum diameter (resp., maximum width) of the sets in the decomposition.

In this note, we study some properties of these minimizing bisections and prove inequalities extending the classical isodiametric and isominwidth inequalities. Furthermore, we address the corresponding reverse optimization problems and establish inequalities similar to the reverse isodiametric and reverse isominwidth inequalities.