Reverse isoperimetric inequality in two-dimensional Alexandrov spaces

Abstract

We prove a reverse isoperimetric inequality for domains homeomorphic to a disc with the boundary of curvature bounded below lying in twodimensional Alexandrov spaces of curvature c. We also study the equality case.Keywords: Alexandrov metric spaces; isoperimetric inequality; λ-convex curve Well-known isoperimetric inequality for the Euclidean plane states that the area F and the length L of the boundary of any plane domain with a rectifiable boundary satisfy the inequalityand equality is attained only for a circle [1]. On the planes of constant curvature there is a similar theorem, and the following inequality holds [2]:where c is the curvature of the plane. In two-dimensional manifolds of bounded curvature for domains homeomorphic to a disc A. D. Alexandrov proved [3] the inequalitywhere ω + is a positive curvature of a domain. In the inequality above equality holds only if the domain is isometric to a lateral surface of a right circular cone with curvature ω + < 2π at the vertex.

Inradius Estimates for Convex Domains in 2-Dimensional Alexandrov Spaces

“…curves whose curvature k, in a weak sense, satisfies k λ > 0 (see Definition 1 below) in constant curvature spaces. Recently, these results were generalized in [Bor2] for λ-convex curves in Alexandrov metric spaces of curvature bounded below. We will use some of the results from [Bor2] in the present paper.…”

“…Recently, these results were generalized in [Bor2] for λ-convex curves in Alexandrov metric spaces of curvature bounded below. We will use some of the results from [Bor2] in the present paper.…”

A sausage body is a unique solution for a reverse isoperimetric problem

“…At the same time, only partial results are currently available for λ-convex bodies. In particular, the two-dimensional case of the reverse isoperimetric problem for λ-convex curves, as we already mentioned in the introduction, is completely solved, see [Bor2,BDr2,BDr3,Dr1]. For higher dimensions the following conjecture is due to Alexander Borisenko (private communication; see also [Dr2,Subsection 4.7]).…”

“…curves whose curvature k, in a weak sense, satisfies k λ > 0. Recently, these results were generalized in [Bor2] for λ-convex curves in Alexandrov metric spaces of curvature bounded below. The result of Borisenko completely settles the reverse isoperimetric problem for λ-convex curves.…”

The estimation of the length of a convex curve in two-dimensional Alexandrov space

“…Such situation is possible only when the curve γ is a plane curve and it is the boundary of a convex cup isometric to the domain G. Recall that the convex cup is a convex surface with a plane boundary γ such that the surface is a graph over a plane domain G enclose by γ. Note that, since γ is a convex curve on the plane, then the integral geodesic curvature of any arc of the curve γ is non-negative viewed both as a curve on the cup and as a curve on a plane [10].…”