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arXiv · 1907.06122

Improved Bounds for Hermite-Hadamard Inequalities in Higher Dimensions

Thomas Beck·Barbara Brandolini·Krzysztof Burdzy·Antoine Henrot·Jeffrey J. Langford·Simon Larson·Robert G. Smits·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 positive, subharmonic function (i.e. $Δf \geq 0$). Then $$ \frac{1}{|Ω|} \int_Ω{f dx} \leq \frac{c_n}{ |\partial Ω| } \int_{\partial Ω}{ f dσ},$$ where $c_n \leq 2n^{3/2}$. This inequality was previously only known for convex functions with a much larger constant. We also show that the optimal constant satisfies $c_n \geq n-1$. As a byproduct, we establish a sharp geometric inequality for two convex domains where one contains the other $ Ω_2 \subset Ω_1 \subset \mathbb{R}^n$: $$ \frac{|\partial Ω_1|}{|Ω_1|} \frac{| Ω_2|}{|\partial Ω_2|} \leq n.$$

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Thomas Beck, Barbara Brandolini, Krzysztof Burdzy, Antoine Henrot, Jeffrey J. Langford, Simon Larson, Robert G. Smits, Stefan Steinerberger. 2019-07-13. Improved Bounds for Hermite-Hadamard Inequalities in Higher Dimensions. https://arxiv.org/abs/1907.06122

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