Pal's isominwidth problem in the hyperbolic space

TL;DR

This paper explores hyperbolic analogs of the classical isominwidth inequality, showing that in hyperbolic space, the minimal volume for convex bodies with fixed width can be arbitrarily small, but in H^2, the regular horocyclic triangle minimizes area.

Contribution

It introduces a hyperbolic version of the isominwidth problem and identifies the minimal area shape in H^2 as the regular horocyclic triangle.

Findings

Volume of convex bodies with fixed Lassak width in H^n can be arbitrarily small.

In H^2, the regular horocyclic triangle minimizes area among bodies with given Lassak width.

Abstract

The paper focuses on possible hyperbolic versions of the classical Pal isominwidth inequality in R^2 from 1921, which states that for a fixed minimal width, the regular triangle has minimal area. We note that the isominwidth problem is still wide open in R^n for n>2. Recent work on the isominwidth problem on the sphere S^2 shows that the solution is the regular spherical triangle when the width is at most \pi/2 according to Bezdek and Blekherman, while Freyer and Sagmeister proved that the minimizer is the polar of a spherical Reuleaux triangle when the minimal width is greater than \pi/2. In this paper, the hyperbolic isominwidth problem is discussed with respect to the probably most natural notion of width due to Lassak in the hyperbolic space H^n where strips bounded by a supporting hyperplane and a corresponding hypersphere are considered. On the one hand, we show that the volume of a convex body of given minimal Lassak width w>0 in H^n might be arbitrarily small; therefore, the isominwidth problem for convex bodies in H^n does not make sense. On the other hand, in the two-dimensional case, we prove that among horocyclically convex bodies of given Lassak width in H^2, the area is minimized by the regular horocyclic triangle.