2021/09/10 by Glaudo, Federico · 2 citations
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2109.04972
We investigate the validity and the stability of various Minkowski-like inequalities for C1-perturbations of the ball. Let K⊆\mathbb 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‖1 d\mathscr Hn-1)^\frac1n-2 ≥ C1(n)Per(K)^\frac1n-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 (∫∂ K H+ d\mathscr Hn-1)^\frac1n-2 ≥ (C1(n)-δ)Per(K)^\frac1n-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 (∫∂ K H+ d\mathscr Hn-1)^\frac1n-2 ≥ C2(n)|K|\frac1n , where C2(n) is the constant that yields the equality when K=B1. 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.