Minkowski inequality for nearly spherical domains

Abstract

We investigate the validity and the stability of various Minkowski-like inequalities for C1-perturbations of the ball. Let K⊆Rn be a domain (possibly not convex and not mean-convex) which is C1-close to a ball. We prove the sharp geometric inequality [Formula presented] where C1(n) is the constant that yields the equality when K=B1 (and ‖II‖1 is the sum of the absolute values of the eigenvalues of the second fundamental form II of ∂K). Moreover, for any δ>0, if K is sufficiently C1-close to a ball, we show the almost sharp Minkowski inequality [Formula presented] If K is axially symmetric, we prove the Minkowski inequality with the sharp constant (i.e., δ=0). We establish also the sharp quantitative stability (in the family of C1-perturbations of the ball) of the volumetric Minkowski inequality [Formula presented] where C2(n) is the constant that yields the equality when K=B1. More precisely, we control the deviation of K from a ball (in a strong norm) with the difference between the left-hand side and the right-hand side of (0.1). Finally, we show, by constructing a counterexample, that the mentioned inequalities are false (even for domains C1-close to the ball) if one replaces H+ with H.