Extreme properties of curves with bounded curvature on a sphere

TL;DR

This paper establishes sharp bounds on the enclosed area of curves on a sphere with bounded geodesic curvature, providing new inequalities for convex curves with curvature constraints.

Contribution

It introduces precise lower and upper bounds for the area of curves on a sphere based on curvature bounds, advancing geometric inequality theory.

Findings

Sharp lower bound on area for curves with bounded geodesic curvature

Dual inequalities for convex curves with curvature bounded above

Enhanced understanding of curvature-area relationships on spheres

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.