SMS scnews item created by Daniel Hauer at Tue 13 Aug 2019 1802
Type: Seminar
Distribution: World
Expiry: 19 Aug 2019
Calendar1: 19 Aug 2019 1200-1300
CalLoc1: AGR Carslaw 829
CalTitle1: On the Hermite-Hadamard formula in higher dimensions
Auth: dhauer@120.17.61.185 (dhauer) in SMS-WASM

# On the Hermite-Hadamard formula in higher dimensions

### Barbara Brandolini

Università degli Studi di Napoli “Federico II”, Italy

## Abstract

Let $\Omega \subset {ℝ}^{n}$ be a convex domain and let $f:\Omega \to ℝ$ be a positive, subharmonic function (i.e. $\Delta f\ge 0$). Then

$\frac{1}{|\Omega |}{\int }_{\Omega }f\phantom{\rule{0.3em}{0ex}}dx\le \frac{{c}_{n}}{|\partial \Omega |}{\int }_{\partial \Omega }f\phantom{\rule{0.3em}{0ex}}d\sigma ,$

where ${c}_{n}\le 2{n}^{3∕2}$. This inequality was previously only known for convex functions with a much larger constant. We also show that the optimal constant satisfies ${c}_{n}\ge n-1$. As a byproduct, we establish the following sharp geometric inequality for two convex domains where one contains the other ${\Omega }_{2}\subset {\Omega }_{1}\subset {ℝ}^{n}$:

$\frac{|\partial {\Omega }_{1}|}{|{\Omega }_{1}|}\frac{|{\Omega }_{2}|}{|\partial {\Omega }_{2}|}\le n.$

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