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math.MG — Metric Geometry

Works in math.MG

5 works
hexagon:2610.00021v1

math.MG — Metric Geometry

A Fourier Consequence for Lattice Kissing Numbers and Average Contacts

Contributed by Scott Kominers

We deduce a new asymptotic upper bound Klat(n)≤(2e/π+o(1))nK_{\mathrm{lat}}(n)\leq\left(\sqrt{2e/\pi}+o(1)\right)^n on the maximal lattice kissing number Klat(n)K_{\mathrm{lat}}(n) in dimension nn, using the explicit auxiliary functions constructed in OpenAI's "Ten Advances" preprint. The same bound holds for the average contact degree of a finite packing of congruent balls. The bound's base-22 exponential rate rounds to 0.39560.3956, matching the value extrapolated empirically by Afkhami-Jeddi, Cohn, Hartman, de Laat, and Tajdini in 2020.

A mix of human-written and AI-generated textHuman understanding: all partsContact numbersDiscrete geometryEuclidean latticesFourier linear programming boundsGeometry of numbersKissing numbersSphere packing
Oct 02, 2026PDF ↗View work →
hexagon:2610.00015v1

math.CO — Combinatorics

Tilings of an equilateral triangle by at most five lattice trapezoids with 60° base angles: a complete structural classification

Contributed by Gonzalo Barria

We study tilings of an equilateral triangle of side n in the triangular grid by k lattice trapezoids with base angles 60°, the objects behind the OEIS sequences A389392 (k = 4) and A391498 (k = 5). We prove two angle identities valid for every such tiling and a lemma relating the number of boundary vertices to the number of pieces having a side on the boundary. With these tools we show that there are exactly 1, 2 and 13 combinatorial types of tilings for k = 3, 4, 5. For k = 3 every tiling is a pinwheel. For k = 4 every tiling belongs to one of the two categories used in A389392, a fact that had previously been taken for granted. For k = 5 the thirteen types refine the eight categories of A391498; in three of them a piece has no side on the boundary. As a consistency check, the volumes of the parameter polytopes together with the generic multiplicities reproduce the leading coefficient 7/36 of the conjectured quasi-polynomial for A391498. All lemmas were checked against an exhaustive enumeration of the tilings with pairwise distinct pieces for n ≤ 15. Finally, the classification turns the thirteen types into eight explicit families of sets of shapes, and an inclusion–exclusion over them reduces the conjectured generating function of A391498 to twelve elementary counting statements, and we settle all twelve: the resulting closed formula reproduces the sequence for every n ≤ 127.

A mix of human-written and AI-generated textHuman understanding: all partsOEIS A391498lattice trapezoidsquasi-polynomialrational generating functiontilings of an equilateral triangle
Oct 01, 2026PDF ↗View work →
hexagon:2610.00011v1

math.MG — Metric Geometry

A generalized Gerver sofa for angled corridors

Contributed by Henrik Schou Guttesen

The moving sofa problem asks for the maximal area of a two-dimensional object capable of being moved through a right-angled corridor of unit width. J.~L.~Gerver constructed a sofa of area 2.2195…2.2195\dots, which was only recently proven to be optimal by J.~Back. Here, I consider the moving sofa problem in an angled corridor, CφC_{\varphi}, of unit width for turn angles 0<φ≤π20 < \varphi \le \frac{\pi}{2}. I first generalize Hammersley's arguments to devise simple analytical lower and upper area bounds for all 0<φ≤π20 < \varphi \le \frac{\pi}{2}. ChatGPT-6 Astra (Codex) is prompted to derive a generalized Gerver sofa parametrized by φ\varphi. The generalized Gerver sofa area is non-analytic at the unique root φ∗∈(0,π/2)\varphi_*\in(0,\pi/2) of exp⁡(φ∗cot⁡φ∗)=1+2cos⁡φ∗−cos⁡2φ∗\exp({\varphi_*\cot\varphi_*}) = 1+2\cos\varphi_*-\cos^2\varphi_*. For turn angles smaller than φ∗\varphi_* a closed-form expression of the sofa area is obtained. The generalized Gerver sofa is conjectured to be the solution to the moving sofa problem for turn angles 0<φ≤π20 < \varphi \le \frac{\pi}{2}.

A mix of human-written and AI-generated textHuman understanding: some partsGeometryGerver-sofamoving-sofa-problemrecreational-math
Oct 01, 2026PDF ↗View work →
hexagon:2609.00146v1

math.MG — Metric Geometry

FINITE HAUSDORFF HYPERSPACES AND GROMOV–HAUSDORFF GEOMETRY

Contributed by Yoshito Ishiki

For a nonempty finite metric space X, let Exp(X) be the space of its nonempty subsets equipped with the Hausdorff metric. We prove that the isometry type of Exp(X) determines X when X is strongly rigid, meaning that distinct unordered pairs of distinct points have dis- tinct distances. We also prove that the hyperspace operation preserves the ordinary Gromov–Hausdorff distance locally under explicit separa- tion conditions on the distance spectra. For finite ultrametric spaces, the hyperspace operation is an isometric embedding with respect to the Gromov–Hausdorff ultrametric. In contrast, we construct finite ultra- metric spaces for which the ordinary Gromov–Hausdorff distance strictly decreases under the hyperspace operation. Examples of equal cardinality exist in every cardinality at least six and realize every ratio in [1/2, 1). Their distances remain constant after the first hyperspace iteration. An explicit perturbation also yields strict contraction for finite strongly rigid metric spaces with all triangle inequalities strict.

Primarily AI-generated textHuman understanding: some partsGromov--Hausdorff spaceHyper space
Sep 30, 2026PDF ↗View work →
hexagon:2609.00123v1

math.MG — Metric Geometry

Breaking the Fourth Wall: The Planar Centroid Banach–Mazur Diameter Is Strictly Less than 44

Contributed by Scott Kominers

We prove that the centroid Banach–Mazur diameter of planar convex bodies is strictly less than 44, (slightly) improving the prior upper bound 69/17≈4.0588269/17\approx 4.05882 due to Lassak. The proof combines a sharp covariance-ellipse sandwich, whose equality cases in any single direction occur only for triangles, with a compactness argument: an extremal pair at distance 44 would consist of two triangles, which are linearly equivalent and hence must in fact be at distance 11. The resulting gap below 44 is uniform but nonquantitative.

A mix of human-written and AI-generated textHuman understanding: all partsBanach–Mazur diameter
Sep 28, 2026PDF ↗View work →