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arXiv · 1905.03216

A Dimension-Free Hermite-Hadamard Inequality via Gradient Estimates for the Torsion Function

Jianfeng Lu·Stefan Steinerberger
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Abstract

Let $Ω\subset \mathbb{R}^n$ be a convex domain and let $f:Ω\rightarrow \mathbb{R}$ be a subharmonic function, $Δf \geq 0$, which satisfies $f \geq 0$ on the boundary $\partial Ω$. Then $$ \int_Ω{f ~dx} \leq |Ω|^{\frac{1}{n}} \int_{\partial Ω}{f ~dσ}.$$ Our proof is based on a new gradient estimate for the torsion function, $Δu = -1$ with Dirichlet boundary conditions, which is of independent interest.

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Jianfeng Lu, Stefan Steinerberger. 2019-05-15. A Dimension-Free Hermite-Hadamard Inequality via Gradient Estimates for the Torsion Function. https://arxiv.org/abs/1905.03216

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