The maximum relative diameter for multi-rotationally symmetric planar convex bodies

Abstract

Abstract. In this work we study the maximum relative diameter functional d M in the class of multi-rotationally symmetric planar convex bodies. A given set C of this class is k -rotationally symmetric for k ∈ {k 1 ,... ,k n } ⊂ N , and so it is natural to consider the standard k i -partition P k i associated to C (which is a minimizing k i -partition for d M when k i 3 ) and the corresponding value d M (P k i ) . We establish the relation among these values, characterizing the particular sets for which all these values coincide.Mathematics subject classification (2010): 52A10, 52A40.

On the isodiametric and isominwidth inequalities for planar bisections

“…Finally, it is worth mentioning that the questions regarding the maximum bisecting diameter (treated firstly in [MPS]) have originated several works in the last years. In [CS] we can find some improvements for the centrally symmetric case, and some related problems for divisions into three or more regions have been studied in [CSS2,C]. Moreover, we also point out that these questions have been partially treated in surfaces of R 3 [CMSS, CSS].…”