The Green–osher Inequality in Relative geometry

Abstract

In this paper we give a proof of the Green–Osher inequality in relative geometry using the minimal convex annulus, including the necessary and sufficient condition for the case of equality.

Nonsymmetric extension of the Green–Osher inequality

“…Using remarkable symmetrization, Gage [4] successfully obtained an inequality for the total squared curvature for convex curves. Following his work, for a planar strictly convex body K and a symmetric, planar strictly convex body E, Green and Osher [8] (see also [12]) obtained a generalized formula:…”

“…Remark 3.5. If R 2 is equipped with a suitable Minkowski metric such that ∂L becomes the isoperimetrix of the Minkowski plane, then (1.4) turns into an inequality in Minkowski geometry (see [12,Remark 3.6]).…”