Extreme Properties of Curves with Bounded Curvature on a Sphere

Abstract

We give a sharp lower bound on the area of the domain enclosed by an embedded curve lying on a two-dimensional sphere, provided that geodesic curvature of this curve is bounded from below. Furthermore, we prove some dual inequalities for convex curves whose curvatures are bounded from above.

Inradius Estimates for Convex Domains in 2-Dimensional Alexandrov Spaces

“…Particularly, Howard and Treibergs [HTr] proved a sharp reverse isoperimetric inequality on the Euclidean plane for closed embedded curves whose curvature k, in a weak sense, satisfies |k| 1, and whose length is in [2π, 14π/3) (see [HTr,Theorem 4.1]). A dual result was obtained in all constant curvature spaces by Borisenko and the author in the series of papers [BDr2,BDr3,Dr1], where a two-dimensional reverse isoperimetric inequality was proved for so-called λ-convex curves, i.e. curves whose curvature k, in a weak sense, satisfies k λ > 0 (see Definition 1 below) in constant curvature spaces.…”

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]).…”

“…Theorem A (and hence Theorem B) for n = 1 and n = 2 was first proven using different techniques in the bachelor thesis of the first author [Ch]. It should be pointed out that Theorem B for n = 1 was suggested earlier in [BDr3]; in that paper the authors also prove a similar result on the two-dimensional sphere.…”

“…At the same time, motivated by the study of strictly convex hypersurfaces in Riemannian spaces (see, for instance, [BM,Bor1,BDr1]), Borisenko and Drach in a series of papers [BDr2,BDr3,Dr1] obtained two-dimensional reverse isoperimetric inequalities for so-called λ-convex curves, i.e. curves whose curvature k, in a weak sense, satisfies k λ > 0.…”

Reverse isoperimetric inequality in two-dimensional Alexandrov spaces

“…For domains in two-dimensional simply-connected spaces of constant curvature equal to c the main theorem was proved when c = 0 in [5], when c = k 2 in [6], and when c = −k 2 in [7]. For J-holomorphic curves some variant of a reverse isoperimetric inequality was proved in [8].…”