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

In this work we study the fencing problem consisting of finnding 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.Comment: Preliminary version, 20 pages, 15 figure

Tight bounds on the maximal perimeter of convex equilateral small polygons

A small polygon is a polygon that has diameter one. The maximal perimeter of a convex equilateral small polygon with $n=2^s$ sides is not known when $s \ge 4$. In this paper, we construct a family of convex equilateral small $n$-gons, $n=2^s$ and $s \ge 4$, and show that their perimeters are within $O(1/n^4)$ of the maximal perimeter and exceed the previously best known values from the literature. In particular, for the first open case $n=16$, our result proves that Mossinghoff's equilateral hexadecagon is suboptimal

Maximal perimeter and maximal width of a convex small polygon

A small polygon is a polygon of unit diameter. The maximal perimeter and the maximal width of a convex small polygon with $n=2^s$ sides are unknown when $s \ge 4$. In this paper, we construct a family of convex small $n$-gons, $n=2^s$ with $s\ge 4$, and show that their perimeters and their widths are within $O(1/n^8)$ and $O(1/n^5)$ of the maximal perimeter and the maximal width, respectively. From this result, it follows that Mossinghoff's conjecture on the diameter graph of a convex small $2^s$-gon with maximal perimeter is not true when $s \ge 4$

Tight bounds on the maximal area of small polygons: Improved Mossinghoff polygons

A small polygon is a polygon of unit diameter. The maximal area of a small polygon with $n=2m$ vertices is not known when $m \ge 7$. In this paper, we construct, for each $n=2m$ and $m\ge 3$, a small $n$-gon whose area is the maximal value of a one-variable function. We show that, for all even $n\ge 6$, the area obtained improves by $O(1/n^5)$ that of the best prior small $n$-gon constructed by Mossinghoff. In particular, for $n=6$, the small $6$-gon constructed has maximal area.Comment: arXiv admin note: text overlap with arXiv:2009.0789

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

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 2 N, k > 3. For general subdivisions, we show that the previous result only holds for k 6 6. We also study the optimal set for this problem, obtaining that for each k 2 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.Ministerio de Educación y Ciencia MTM2010-21206-C02-01Ministerio de Economía e Innovación MTM2013-48371-C2-1-PJunta de Andalucía FQM-325Junta de Andalucía P09-FQM-508

On the geometric dilation of closed curves, graphs, and point sets

The detour between two points u and v (on edges or vertices) of an embedded planar graph whose edges are curves is the ratio between the shortest path in in the graph between u and v and their Euclidean distance. The maximum detour over all pairs of points is called the geometric dilation. Ebbers-Baumann, Gruene and Klein have shown that every finite point set is contained in a planar graph whose geometric dilation is at most 1.678, and some point sets require graphs with dilation at least pi/2 = 1.57... We prove a stronger lower bound of 1.00000000001*pi/2 by relating graphs with small dilation to a problem of packing and covering the plane by circular disks. The proof relies on halving pairs, pairs of points dividing a given closed curve C in two parts of equal length, and their minimum and maximum distances h and H. Additionally, we analyze curves of constant halving distance (h=H), examine the relation of h to other geometric quantities and prove some new dilation bounds.Comment: 31 pages, 16 figures. The new version is the extended journal submission; it includes additional material from a conference submission (ref. [6] in the paper

On the Lengths of Curves Passing through Boundary Points of a Planar Convex Shape

We study the lengths of curves passing through a fixed number of points on the boundary of a convex shape in the plane. We show that for any convex shape $K$, there exist four points on the boundary of $K$ such that the length of any curve passing through these points is at least half of the perimeter of $K$. It is also shown that the same statement does not remain valid with the additional constraint that the points are extreme points of $K$. Moreover, the factor $\frac12$ cannot be achieved with any fixed number of extreme points. We conclude the paper with few other inequalities related to the perimeter of a convex shape.Comment: 7 pages, 8 figure