$$\ell $$-convex Legendre curves and geometric inequalities

Abstract

AbstractIn this paper we consider $$\ell $$ ℓ -convex Legendre curves, which are natural generalizations of strictly convex curves. We generalize various optimal geometric inequalities, isoperimetric inequality, Bonnesen’s inequality and Green–Osher inequality, for strictly convex curves to ones for $$\ell $$ ℓ -convex Legendre curves. Moreover we generalize the inverse curvature curve flow for this class of Legendre curves and prove that it always converges to a compact soliton after rescaling. Unlike in the class of regular curves, there are infinitely many compact solitons, which include circles and astroids.