  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.
