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The Hermite–Hadamard Inequality in Higher Dimensions

  • Stefan SteinerbergerEmail author
Article
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Abstract

The Hermite–Hadamard inequality states that the average value of a convex function on an interval is bounded from above by the average value of the function at the endpoints of the interval. We provide a generalization to higher dimensions: let \(\Omega \subset {\mathbb {R}}^n\) be a convex domain and let \(f:\Omega \rightarrow {\mathbb {R}}\) be a convex function satisfying \(f \big |_{\partial \Omega } \ge 0\), then
$$\begin{aligned} \frac{1}{|\Omega |} \int _{\Omega }{f ~\text {d} {\mathcal {H}}^n} \le \frac{2 \pi ^{-1/2} n^{n+1}}{|\partial \Omega |} \int _{\partial \Omega }{f~\text {d} {\mathcal {H}}^{n-1}}. \end{aligned}$$
The constant \(2 \pi ^{-1/2} n^{n+1}\) is presumably far from optimal; however, it cannot be replaced by 1 in general. We prove slightly stronger estimates for the constant in two dimensions where we show that \(9/8 \le c_2 \le 8\). We also show, for some universal constant \(c>0\), if \(\Omega \subset {\mathbb {R}}^2\) is simply connected with smooth boundary, \(f:\Omega \rightarrow {\mathbb {R}}_{}\) is subharmonic, i.e., \(\Delta f \ge 0\), and \(f \big |_{\partial \Omega } \ge 0\), then
$$\begin{aligned} \int _{\Omega }{f~ \text {d} {\mathcal {H}}^2} \le c \cdot \text{ inradius }(\Omega ) \int _{\partial \Omega }{ f ~\text {d}{\mathcal {H}}^{1}}. \end{aligned}$$
We also prove that every domain \(\Omega \subset {\mathbb {R}}^n\) whose boundary is ’flat’ at a certain scale \(\delta \) admits a Hermite–Hadamard inequality for all subharmonic functions with a constant depending only on the dimension, the measure \(|\Omega |\), and the scale \(\delta \).

Keywords

Hermite–Hadamard inequality Subharmonic functions Brownian motion 

Mathematics Subject Classification

26B25 28A75 31A05 31B05 35B50 

Notes

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Copyright information

© Mathematica Josephina, Inc. 2019

Authors and Affiliations

  1. 1.Department of MathematicsYale UniversityNew HavenUSA

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