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(∫∂K‖II‖1dHn−1)1n−2≥C1(n)Per(K)1n−1, 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(∫∂KH+dHn−1)1n−2≥(C1(n)−δ)Per(K)1n−1. 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(0.1)(∫∂KH+dHn−1)1n−2≥C2(n)|K|1n, 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.