Reverse Isoperimetric Properties of Thick $\lambda$-Concave Bodies in the Hyperbolic Plane

TL;DR

This paper proves that in the hyperbolic plane, the thick λ-sausage shape uniquely minimizes area among convex bodies with bounded curvature, extending reverse isoperimetric inequalities to hyperbolic geometry.

Contribution

It establishes the minimality of the thick λ-sausage body for area under curvature constraints in hyperbolic space, introducing new techniques to handle non-convex inner parallel bodies.

Findings

Thick λ-sausage bodies are the unique area minimizers under given boundary length and curvature bounds.

The study extends reverse isoperimetric inequalities to hyperbolic geometry with curvature constraints.

A new approach handles non-convex inner parallel bodies in hyperbolic space.

Abstract

In this paper we address the reverse isoperimetric inequality for convex bodies with uniform curvature constraints in the hyperbolic plane $\mathbb{H}^2$. We prove that the\textit{ thick $\lambda$-sausage} body, that is, the convex domain bounded by two equal circular arcs of curvature $\lambda$ and two equal arcs of hypercircle of curvature $1 / \lambda$, is the unique minimizer of area among all bodies $K \subset \mathbb{H}^2$ with a given length and with curvature of $\partial K$ satisfying $1 / \lambda \leq \kappa \leq \lambda$ (in a weak sense). We call this class of bodies \textit{thick $\lambda$-concave} bodies, in analogy to the Euclidean case where a body is $\lambda$-concave if $0 \leq \kappa \leq \lambda$. The main difficulty in the hyperbolic setting is that the inner parallel bodies of a convex body are not necessarily convex. To overcome this difficulty, we introduce an extra assumption of thickness $\kappa \geq 1/\lambda$.