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

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 $1/\lambda \le \kappa \le \lambda$ (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 $0\le \kappa \le \lambda$. 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 $\kappa \ge 1/\lambda$.

14 pages, 7 figures