Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Maria Esteban

TL;DR

The paper solves a reverse isoperimetric problem for thick $\lambda$-concave bodies in the hyperbolic plane $\mathbb{H}^2$, identifying the thick $\lambda$-sausage as the unique area minimizer at fixed boundary length under curvature bounds $\tfrac{1}{\lambda} \le \kappa \le \lambda$. The authors develop a hyperbolic Steiner framework and address the non-convexity of inner parallel bodies by leveraging a thickness condition, proving both the minimization result and a sharp lower bound on area that recovers the Euclidean case as curvature tends to zero. A key methodological contribution is a Blaschke-type rolling theorem for thick $\lambda$-concave bodies in $\mathbb{H}^2$, establishing that a ball of curvature $\lambda$ can freely roll inside the body under the thickness assumption, with optimality of this condition. The work bridges Euclidean reverse isoperimetric results with hyperbolic geometry, offering precise characterizations and enabling a robust comparison principle via inner parallel transformations and hyperbolic Steiner-type formulas. These results deepen understanding of convex geometry in spaces of constant negative curvature and have potential implications for geometric optimization under curvature constraints in non-Euclidean settings.

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 $λ$-sausage} body, that is, the convex domain bounded by two equal circular arcs of curvature $λ$ and two equal arcs of hypercircle of curvature $1 / λ$, 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 $1 / λ\leq κ\leq λ$ (in a weak sense). We call this class of bodies \textit{thick $λ$-concave} bodies, in analogy to the Euclidean case where a body is $λ$-concave if $0 \leq κ\leq λ$. 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 $κ\geq 1/λ$.

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

TL;DR

The paper solves a reverse isoperimetric problem for thick -concave bodies in the hyperbolic plane , identifying the thick -sausage as the unique area minimizer at fixed boundary length under curvature bounds . The authors develop a hyperbolic Steiner framework and address the non-convexity of inner parallel bodies by leveraging a thickness condition, proving both the minimization result and a sharp lower bound on area that recovers the Euclidean case as curvature tends to zero. A key methodological contribution is a Blaschke-type rolling theorem for thick -concave bodies in , establishing that a ball of curvature can freely roll inside the body under the thickness assumption, with optimality of this condition. The work bridges Euclidean reverse isoperimetric results with hyperbolic geometry, offering precise characterizations and enabling a robust comparison principle via inner parallel transformations and hyperbolic Steiner-type formulas. These results deepen understanding of convex geometry in spaces of constant negative curvature and have potential implications for geometric optimization under curvature constraints in non-Euclidean settings.

Abstract

In this paper we address the reverse isoperimetric inequality for convex bodies with uniform curvature constraints in the hyperbolic plane . We prove that the\textit{ thick -sausage} body, that is, the convex domain bounded by two equal circular arcs of curvature and two equal arcs of hypercircle of curvature , is the unique minimizer of area among all bodies with a given length and with curvature of satisfying (in a weak sense). We call this class of bodies \textit{thick -concave} bodies, in analogy to the Euclidean case where a body is -concave if . 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 .

Paper Structure

This paper contains 11 sections, 11 theorems, 23 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Some reverse isoperimetric results
  3. Reverse isoperimetric inequality in $\mathbb{H}^2$
  4. Structure of the paper
  5. Acknowledgements
  6. Background Theory
  7. Curves of constant curvature in $\mathbb{H}^2$
  8. Some useful lemmas
  9. Proof of Theorems \ref{['th1']} and \ref{['th2']}
  10. Construction of the inner parallel bodies
  11. Steiner's formula in the Hyperbolic Space

Key Result

Theorem 1.1

(kostya3, see also saorinnayar). Let $n \geq 2$. Let $K \subset \mathbb{R}^n$ be a $\lambda$-concave body, and let $S \subset \mathbb{R}^n$ be a $\lambda$-sausage, i.e., the convex hull of two balls of radius $1/\lambda$. If $|\partial K| = |\partial S|$, then $|K| \geq |S|$, and equality holds if a

Figures (7)

  • Figure 1: $\lambda$-sausage (left) and $\lambda$-convex lens (right) kostya3
  • Figure 2: Thick $\lambda$-sausage
  • Figure 3: Circle (green), Geodesics (orange), Horocycles (blue) and Hypercircles (pink)
  • Figure 4: The proof of Lemma \ref{['lastlemma']} is constructed as follows. We consider the triangle formed by the center of the unit disk (black), the center of the hypercircle (green), and their point of intersection. By examining the properties of the tangent lines to the disk and hypercircle at the point of intersection, we can establish a relationship between the radius $R$, the curvature at (0,0) and $\beta$.
  • Figure 5: Hypercircle in the Upper Half-Plane
  • ...and 2 more figures

Theorems & Definitions (27)

  • Definition 1.1: $\lambda$-convex and $\lambda$-concave bodies in $\mathbb{H}^2$ kostya
  • Definition 1.2: Thick $\lambda$-concave bodies
  • Theorem 1.1: Reverse Isoperimetric Inequality for $\lambda$-concave Bodies
  • Definition 1.3: Thick $\lambda$-sausage
  • Theorem 1.2: Reverse isoperimetric inequality for thick $\lambda$-concave bodies in $\mathbb{H}^2$
  • Theorem 1.3: Lower bound for reverse isoperimetric inequality for thick $\lambda$-concave bodies in $\mathbb{H}^2$
  • Remark 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 17 more