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.

Bisections of Centrally Symmetric Planar Convex Bodies Minimizing the Maximum Relative Diameter

“…For a fixed centrally symmetric planar convex body C, the purpose of these notes is investigating the bisections of C that minimize the maximum relative diameter functional, in the same spirit as in [4,5], see also [11]: determining these bisections precisely or, at least, describing some of their geometrical properties. These bisections will be called minimizing along this paper.…”

“…As indicated in the Introduction, it is known [5,Th. 4.5] that, for any k-rotationally symmetric planar convex body, its corresponding standard kpartition (defined by means of k inradius segments symmetrically placed) is always minimizing for the maximum relative diameter functional, when k 3, p p Figure 8.…”

“…In this context, we can investigate the k-partitions of C attaining the minimum value for d M . This was treated in [5] (see also [4]), where it is proved that the so-called standard k-partition (constructed by using k inradius segments symmetrically placed, see Figure 1) is a solution for this problem, for any k 3. We shall see in Section 4 that the previous result does not hold in our setting (which would correspond to k = 2): a standard bisection (consisting of two symmetric inradius segments) is not minimizing in general (for instance, see Example 4.2).…”

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

“…The following definition describes the decompositions we shall consider for multi-rotationally symmetric planar convex bodies. Since this kind of sets have a special interior point (which is the center of symmetry), it is natural, in some sense, working with a particular type of divisions called k-partitions, where k ∈ N, see [3]. Definition 2.9.…”

“…The above problem involving the maximum relative diameter was studied partially for k = 2 in [9], proving that a minimizing decomposition into two equal-area connected subsets is given by a straight line passing through the center of symmetry of the set [9,Prop.4], but a complete and more precise characterization is not known yet. Later on, for k 3, the general problem was treated in [3], see also [2], obtaining that the so-called standard k-partition is a minimizing k-partition (being also a minimizing decomposition without additional restrictions when k 6) [3,Th. 4.5 and 4.6].…”