Trisections of a 3-rotationally symmetric planar convex body minimizing the maximum relative diameter

Abstract

In this work we study the fencing problem consisting of finding a trisection of a 3-rotationally symmetric planar convex body which minimizes the maximum relative diameter. We prove that an optimal solution is given by the so-called standard trisection. We also determine the optimal set giving the minimum value for this functional and study the corresponding universal lower bound.

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

“…4.7]. We remark that, although the results in [3] are stated for divisions into three subsets of equal areas, all of them also hold in the case of unequal areas. This paper is inspired precisely in these last two references involving the maximum relative diameter [11,3].…”

“…In this setting, the partitioning problem seeks the decomposition of C into k subsets with the least possible value for the maximum relative diameter. This problem has been recently studied for k = 2 and k = 3 under some additional symmetry hypotheses [11,3], and the results therein constitute the main motivation for our work.…”

“…5.1] for any set of that class. In addition, the optimal set for this problem is also characterized, up to dilations, as the intersection of the unit circle with a certain equilateral triangle [3,Th. 4.7].…”

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

“…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 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].…”