Subdivisions of rotationally symmetric planar convex bodies minimizing the maximum relative diameter

Abstract

In this work we study subdivisions of k-rotationally symmetric planar convex bodies that minimize the maximum relative diameter functional. For some particular subdivisions called k-partitions, consisting of k curves meeting in an interior vertex, we prove that the so-called standard k-partition (given by k equiangular inradius segments) is minimizing for any k∈N, k⩾3. For general subdivisions, we show that the previous result only holds for k⩽6. We also study the optimal set for this problem, obtaining that for each k∈N, k⩾3, it consists of the intersection of the unit circle with the corresponding regular k-gon of certain area. Finally, we also discuss the problem for planar convex sets and large values of k, and conjecture the optimal k-subdivision in this case.