On the isodiametric and isominwidth inequalities for planar bisections

Abstract

For a given planar convex compact set $K$, consider a bisection $\{A,B\}$ of $K$ (i.e., $A\cup B=K$ and whose common boundary $A\cap B$ is an injective continuous curve connecting two boundary points of $K$) minimizing the corresponding maximum diameter (or maximum width) of the regions among all such bisections of $K$. In this note we study some properties of these minimizing bisections and we provide analogous to the isodiametric (Bieberbach, 1915), the isominwidth (P\'al, 1921), the reverse isodiametric (Behrend, 1937), and the reverse isominwidth (Gonz\'alez Merino \& Schymura, 2018) inequalities.