Let $\Omega \subset \mathbb{R}^d$, $d \geq 2$, be a bounded convex domain and $f\colon \Omega \to \mathbb{R}$ be a non-negative subharmonic function. In this paper we prove the inequality \[ \frac{1}{|\Omega|}\int_\Omega f(x)\,dx \leq \frac{d}{|\partial\Omega|}\int_{\partial\Omega} f(x)\,d\sigma(x)\,. \] Equivalently, the result can be stated as a bound for the gradient of the Saint Venant torsion function. Specifically, if $\Omega \subset \mathbb{R}^d$ is a bounded convex domain and $u$ is the solution of $-\Delta u =1$ with homogeneous Dirichlet boundary conditions, then \[ \|\nabla u\|_{L^\infty(\Omega)} < d\frac{|\Omega|}{|\partial\Omega|}\,. \] Moreover, both inequalities are sharp in the sense that if the constant $d$ is replaced by something smaller there exist convex domains for which the inequalities fail. This improves upon the recent result that the optimal constant is bounded from above by $d^{3/2}$ due to Beck et al.

中文翻译：

---

一个尖锐的多维 Hermite-Hadamard 不等式

令 $\Omega \subset \mathbb{R}^d$, $d \geq 2$ 是一个有界凸域，$f\colon \Omega \to \mathbb{R}$ 是一个非负次谐波函数。本文证明不等式\[ \frac{1}{|\Omega|}\int_\Omega f(x)\,dx \leq \frac{d}{|\partial\Omega|}\int_{\部分\Omega} f(x)\,d\sigma(x)\,. \] 等效地，结果可以表示为圣维南扭转函数梯度的界限。具体来说，如果 $\Omega \subset \mathbb{R}^d$ 是一个有界凸域，而 $u$ 是 $-\Delta u =1$ 在齐次狄利克雷边界条件下的解，那么 \[ \|\nabla u\|_{L^\infty(\Omega)} < d\frac{|\Omega|}{|\partial\Omega|}\,. \] 此外，这两个不等式都是尖锐的，因为如果常数 $d$ 被较小的值替换，则存在不等式失败的凸域。