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Minkowski inequality for nearly spherical domains

We investigate the validity and the stability of various Minkowski-like inequalities for $C^1$-perturbations of the ball. Let $K\subseteq\mathbb R^n$ be a domain (possibly not convex and not mean-convex) which is $C^1$-close to a ball. We prove the sharp geometric inequality $$ \left(\int_{\partial K} \lVert II\rVert_1 d\mathscr H^{n-1}\right)^{\frac1{n-2}} \ge C_1(n)Per(K)^{\frac1{n-1}} , $$ where $C_1(n)$ is the constant that yields the equality when $K=B_1$ (and $\lVert II\rVert_1$ is the sum of the absolute values of the eigenvalues of the second fundamental form $II$ of $\partial K$). Moreover, for any $δ>0$, if $K$ is sufficiently $C^1$-close to a ball, we show the almost sharp Minkowski inequality $$ \left(\int_{\partial K} H^+ d\mathscr H^{n-1}\right)^{\frac1{n-2}} \ge (C_1(n)-δ)Per(K)^{\frac1{n-1}} . $$ 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 $C^1$-perturbations of the ball) of the volumetric Minkowski inequality $$ \left(\int_{\partial K} H^+ d\mathscr H^{n-1}\right)^{\frac1{n-2}} \ge C_2(n)|K|^{\frac1n} , $$ where $C_2(n)$ is the constant that yields the equality when $K=B_1$. Finally, we show, by constructing a counterexample, that the mentioned inequalities are false (even for domains $C^1$-close to the ball) if one replaces $H^+$ with $H$.