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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 injectivecontinuous curve connecting two boundary points of $K$) minimizing thecorresponding maximum diameter (or maximum width) of the regions among all suchbisections of $K$.  In this note we study some properties of these minimizing bisections and weprovide analogous to the isodiametric (Bieberbach, 1915), the isominwidth(P\'al, 1921), the reverse isodiametric (Behrend, 1937), and the reverseisominwidth (Gonz\'alez Merino \& Schymura, 2018) inequalities.

On the isodiametric and isominwidth inequalities for planar bisections