A Sharp Multidimensional Hermite–Hadamard Inequality

Abstract

Let Ω ⊂ R^d, d ≥ 2, be a bounded convex domain and f: Ω → R be a non-negative subharmonic function. In this paper, we prove the inequality 1/|Ω| ∫_Ω f(x)dx ≤ d/|∂Ω| ∫_(∂Ω) f(x) dσ(x). Equivalently, the result can be stated as a bound for the gradient of the Saint Venant torsion function. Specifically, if Ω ⊂R^d is a bounded convex domain and u is the solution of -Δu =1 with homogeneous Dirichlet boundary conditions, then ||∇u‖_((L^∞)(Ω)) < d|Ω|/|∂Ω|. 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. [2].

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