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1. The Limit Point of the Pentagram Map and Infinitesimal Monodromy

   https://doi.org/10.1093/imrn/rnaa258  

   Aboud, Quinton ; Izosimov, Anton ( September 2020 , 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. 

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2. Maximal Distortion of Geodesic Diameters in Polygonal Domains

   Adrian Dumitrescu ; Csaba D. Tóth ( June 2023 , 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→∞ . 

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3. Maximal Distortion of Geodesic Diameters in Polygonal Domains

   Dumitrescu, Adrian ; Toth, Csaba D. ( June 2023 , 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→∞ 

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4. Maximal Distortion of Geodesic Diameters in Polygonal Domains

   Adrian Dumitrescu ; Csaba D. Tóth ( June 2023 , 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 → ∞ 

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5. Balanced centroidal power diagrams for redistricting

   https://doi.org/10.1145/3274895.3274979  

   Cohen-Addad, Vincent ; Klein, Philip N. ; Young, Neal E. ( November 2018 , 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. 

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