Improved bounds for Hermite-Hadamard inequalities in higher dimensions

Barbara Brandolini, Barbara Brandolini, Stefan Steinerberger, Jeffrey J. Langford, Simon Larson, Thomas Beck, Stefan Steinerberger, Antoine Henrot, Krzysztof Burdzy, Robert Smits

Research output: Contribution to journalArticlepeer-review


Let $Omega subset mathbb{R}^n$ be a convex domain and let $f:Omega ightarrow mathbb{R}$ be a positive, subharmonic function (i.e. $Delta f geq 0$). Then$$ rac{1}{|Omega|} int_{Omega}{f dx} leq rac{c_n}{ |partial Omega| } int_{partial Omega}{ f dsigma},$$where $c_n leq 2n^{3/2}$. This inequality was previously only known for convex functions with a much larger constant. We alsoshow 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 $ Omega_2 subset Omega_1 subset mathbb{R}^n$: $$ rac{|partial Omega_1|}{|Omega_1|} rac{| Omega_2|}{|partial Omega_2|} leq n.$$
Original languageEnglish
Pages (from-to)801-816
Number of pages16
Publication statusPublished - 2019

All Science Journal Classification (ASJC) codes

  • Geometry and Topology


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