# On the isodiametric and isominwidth inequalities for planar bisections

## TL;DR

This paper investigates optimal planar bisections of convex sets that minimize maximum diameter or width, and establishes related inequalities analogous to classical isodiametric and isominwidth results, including their reverse forms.

## Contribution

It introduces properties of minimal bisections in convex sets and derives new inequalities similar to classical geometric inequalities, extending the theory to reverse cases.

## Key findings

- Properties of minimizing bisections analyzed
- Analogous isodiametric and isominwidth inequalities established
- Reverse inequalities for these measures derived

## 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.