Title: The pentagram map, Poncelet polygons, and commuting difference operators

The pentagram map takes a planar polygon $P$ to a polygon $P'$ whose vertices are the intersection points of consecutive shortest diagonals of $P$ . This map is known to interact nicely with Poncelet polygons, that is, polygons which are simultaneously inscribed in a conic and circumscribed about a conic. A theorem of Schwartz states that if $P$ is a Poncelet polygon, then the image of $P$ under the pentagram map is projectively equivalent to $P$ . In the present paper, we show that in the convex case this property characterizes Poncelet polygons: if a convex polygon is projectively equivalent to its pentagram image, then it is Poncelet. The proof is based on the theory of commuting difference operators, as well as on properties of real elliptic curves and theta functions. more »« less

Aboud, Quinton; Izosimov, Anton(
, International Mathematics Research Notices)

null
(Ed.)

Abstract The pentagram map takes a planar polygon $P$ to a polygon $P^{\prime }$ whose vertices are the intersection points of the consecutive shortest diagonals of $P$. The orbit of a convex polygon under this map is a sequence of polygons that converges exponentially to a point. Furthermore, as recently proved by Glick, coordinates of that limit point can be computed as an eigenvector of a certain operator associated with the polygon. In the present paper, we show that Glick’s operator can be interpreted as the infinitesimal monodromy of the polygon. Namely, there exists a certain natural infinitesimal perturbation of a polygon, which is again a polygon but in general not closed; what Glick’s operator measures is the extent to which this perturbed polygon does not close up.

Adrian Dumitrescu; Csaba D. Tóth(
, Proc. 34th International Workshop on Combinatorial Algorithms)

For a polygon P with holes in the plane, we denote by ρ(P ) the ratio between the geodesic and the Euclidean diameters of P . It is shown that over all convex polygons with h convex holes, the supremum of ρ(P ) is between Ω(h1/3) and O(h1/2). The upper bound improves
to O(1 + min{h3/4∆, h1/2∆1/2}) if every hole has diameter at most ∆ ·diam2(P ); and to O(1) if every hole is a fat convex polygon. Furthermore, we show that the function g(h) = supP ρ(P ) over convex polygons with h convex holes has the same growth rate as an analogous quantity over geometric triangulations with h vertices when h → ∞

For a polygon P with holes in the plane, we denote by ϱ(P)
the ratio between the geodesic and the Euclidean diameters of P. It is shown that over all convex polygons with h convex holes, the supremum of ϱ(P)
is between Ω(h1/3)
and O(h1/2)
. The upper bound improves to O(1+min{h3/4Δ,h1/2Δ1/2})
if every hole has diameter at most Δ⋅diam2(P)
; and to O(1) if every hole is a fat convex polygon. Furthermore, we show that the function g(h)=supPϱ(P)
over convex polygons with h convex holes has the same growth rate as an analogous quantity over geometric triangulations with h vertices when h→∞

Adrian Dumitrescu; Csaba D. Tóth(
, Combinatorial Algorithms (IWOCA 2023))

For a polygon P with holes in the plane, we denote by ϱ(P) the ratio between the geodesic and the Euclidean diameters of P. It is shown that over all convex polygons with h convex holes, the supremum of ϱ(P) is between Ω(h1/3) and O(h1/2) . The upper bound improves to O(1+min{h3/4Δ,h1/2Δ1/2}) if every hole has diameter at most Δ⋅diam2(P) ; and to O(1) if every hole is a fat convex polygon. Furthermore, we show that the function g(h)=supPϱ(P)
over convex polygons with h convex holes has the same growth rate as an analogous quantity over geometric triangulations with h vertices when h→∞
.

Cohen-Addad, Vincent; Klein, Philip N.; Young, Neal E.(
, Proceedings of the 26th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems)

We consider the problem of political redistricting: given the locations of people in a geographical area (e.g. a US state), the goal is to decompose the area into subareas, called districts, so that the populations of the districts are as close as possible and the districts are ``compact'' and ``contiguous,'' to use the terms referred to in most US state constitutions and/or US Supreme Court rulings.
We study a method that outputs a solution in which each district is the intersection of a convex polygon with the geographical area. The average number of sides per polygon is less than six. The polygons tend to be quite compact. Every two districts differ in population by at most one (so we call the solution balanced).
In fact, the solution is a centroidal power diagram: each polygon has an associated center in ℝ³ such that
* the projection of the center onto the plane z = 0 is the centroid of the locations of people assigned to the polygon, and
* for each person assigned to that polygon, the polygon's center is closest among all centers. The polygons are convex because they are the intersections of 3D Voronoi cells with the plane.
The solution is, in a well-defined sense, a locally optimal solution to the problem of choosing centers in the plane and choosing an assignment of people to those 2-d centers so as to minimize the sum of squared distances subject to the assignment being balanced.
* A practical problem with this approach is that, in real-world redistricting, exact locations of people are unknown. Instead, the input consists of polygons (census blocks) and associated populations. A real redistricting must not split census blocks. We therefore propose a second phase that perturbs the solution slightly so it does not split census blocks. In our experiments, the second phase achieves this while preserving perfect population balance. The district polygons are no longer convex at the fine scale because their boundaries must follow the boundaries of census blocks, but at a coarse scale they preserve the shape of the original polygons.

@article{osti_10413208,
place = {Country unknown/Code not available},
title = {The pentagram map, Poncelet polygons, and commuting difference operators},
url = {https://par.nsf.gov/biblio/10413208},
DOI = {10.1112/S0010437X22007345},
abstractNote = {The pentagram map takes a planar polygon $P$ to a polygon $P'$ whose vertices are the intersection points of consecutive shortest diagonals of $P$ . This map is known to interact nicely with Poncelet polygons, that is, polygons which are simultaneously inscribed in a conic and circumscribed about a conic. A theorem of Schwartz states that if $P$ is a Poncelet polygon, then the image of $P$ under the pentagram map is projectively equivalent to $P$ . In the present paper, we show that in the convex case this property characterizes Poncelet polygons: if a convex polygon is projectively equivalent to its pentagram image, then it is Poncelet. The proof is based on the theory of commuting difference operators, as well as on properties of real elliptic curves and theta functions.},
journal = {Compositio Mathematica},
volume = {158},
number = {5},
author = {Izosimov, Anton},
}

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