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Ádám Sagmeister

Publications and source records attributed to Ádám Sagmeister.

11 recordsLinked to original sources
arXiv2026

Optimal stability of Pál's isominwidth inequality for ball convex bodies in planes of constant curvature

Pál's isominwidth inequality (1921) answered the Kakeya needle problem (1917) for convex sets. It states that among convex bodies of fixed minimum width $w$ in the Euclidean plane, the regular triangle has minimal area. The isominwidth inequality was generalized to the $2$-dimensional sphere by Bezdek and Blekherman and Freyer and Sagmeister (arXiv:2411.11462). Interestingly, in hyperbolic space, no minimizer exists, as shown by Böröczky, Freyer and Sagmeister (arXiv:2502.04427). The stability of the Euclidean Pál inequality with respect to the Hausdorff metric and the symmetric difference metric was proved by Lucardesi and Zucco (arXiv:2405.18294). Fodor, Robock and Sagmeister (arXiv:2602.19300) proved $r$-ball convex analogs of the isominwidth inequality in all three constant curvature planes connecting Pál's theorem with the Blaschke--Lebesgue inequality. In this paper, we prove optimal stability versions of this statement with respect to the Hausdorff distance and the symmetric difference metric in all three constant curvature planes.

math.MG↗
arXiv2026

Random convergence of generalized Steiner processes

It is well-known that a random Steiner process converges almost surely to a Euclidean ball. On the other hand, as we show in this article, a planar random shaking process converges almost surely to a triangle (in the Banach-Mazur distance). Asking whether these phenomena are generic or exceptional, we study random iterations of an operation called $λ$-Steiner symmetrization, which includes Steiner symmetrization and shaking as special cases. We establish abstract criteria for almost sure convergence and divergence of such processes, obtaining criteria for many other symmetrization operations from convex geometry as a byproduct. Based on a joint characterization of ellipses and triangles, we give a full resolution to our question in the planar case.

math.MG↗
arXiv2026

Pál's isominwidth inequality for ball convex bodies in planes of constant curvature

Pál's classical isominwidth inequality states that the regular triangle has minimal area among plane convex bodies of minimal width $w$. A similar result is the Blaschke--Lebesgue inequality that states that Reuleaux triangles minimize the area among bodies of constant width $w$ in the plane. In this paper, we connect these two problems by solving the isominwidth problem for $r$-ball convex bodies in the Euclidean, hyperbolic and spherical planes.

math.MG↗
arXiv2025

Penny graphs in the hyperbolic plane

We consider the problem of finding the maximum number $e_d(n)$ of pairs of touching circles in a packing of $n$ congruent circles of diameter $d$ in the hyperbolic plane of curvature $-1$. In the Euclidean plane, the maximum comes from a spiral construction of the tiling of the plane with equilateral triangles (Harborth 1974), with a similar result in the hyperbolic plane for the values of $d$ corresponding to the order-$k$ triangular tilings (Bowen 2000). We present various upper and lower bounds for $e_d(n)$ for all values of $d > 0$. In particular, we prove that if $d > 0.66114\dots$ except for $d=0.76217\dots$, then the number of touching pairs is less than the one coming from a spiral construction in the order-$7$ triangular tiling, which we conjecture to be extremal. We also give a lower bound $e_d(n) > (2+\varepsilon_d)n$ where $\varepsilon_d > 1$ for all $d > 0$.

math.CO↗
arXiv2025

John Ellipsoids of Revolution

Finding a largest Euclidean ball in a given convex body $K \subset \mathbb{R}^d$ and finding a largest volume ellipsoid in $K$ are two problems of fundamentally different nature. The first is a purely Euclidean problem, where we consider scaled copies of the origin-centered closed unit ball, whereas in the second problem, we search among all affine copies of the unit ball. In this paper, we interpolate between these two classical problems by considering ellipsoids of revolution. More generally, we study pairs of convex bodies $K$ and $L$, and seek a largest-volume affine image of $K$ contained within $L$, subject to certain restrictions on the allowed affine transformations. We derive first-order necessary conditions for optimality, generalizing known conditions from the unrestricted affine setting. Using these conditions, we show that an extremal ellipsoid of revolution exhibits properties analogous to those of either the largest-volume ellipsoid or the largest Euclidean ball, depending on whether the ellipsoid is considered along its axis of revolution or along the orthogonal complement of that axis.

math.MG↗
arXiv2024

The isominwidth problem on the 2-sphere

Pál's isominwidth theorem states that for a fixed minimal width, the regular triangle has minimal area. A spherical version of this theorem was proven by Bezdek and Blekherman, if the minimal width is at most $\tfrac π2$. If the width is greater than $\tfrac π2$, the regular triangle no longer minimizes the area at fixed minimal width. We show that the minimizers are instead given by the polar sets of spherical Reuleaux triangles. Moreover, stability versions of the two spherical inequalities are obtained.

math.MG↗
arXiv2024

On the area of ordinary hyperbolic reduced polygons

A convex body $R$ in the hyperbolic plane is reduced if any convex body $K\subset R$ has a smaller minimal width than $R$. We examine the area of a family of hyperbolic reduced $n$-gons, and prove that, within this family, regular $n$-gons have maximal area.

math.MG↗