Beck, Thomas; Brandolini, Barbara; Burdzy, Krzysztof; Henrot, Antoine; Langford, Jeffrey J.; Larson, Simon; Smits, Robert; Steinerberger, Stefan Improved bounds for Hermite-Hadamard inequalities in higher dimensions. (English) Zbl 1462.26022 J. Geom. Anal. 31, No. 1, 801-816 (2021). In this paper, it iss proved that: If \(\Omega \subset \mathbb{R}^n\) is a convex domain and \(f: \Omega \to \mathbb{R}\) is positive and subharmonic function, then \[\frac1{\mid \Omega \mid}\int_{\Omega} f(x)dx \le \frac{c_n}{\mid \partial \Omega\mid}\int_{\partial \Omega} f\sigma d\sigma,\] where \(c_n \le 2n^{3/2}\). This inequality has been known for convex functions with much larger constant, however in this paper it was shown that the optimal constant satisfies \(c_n \ge n-1.\) As a consequence of the above result, the authors established a sharp geometric inequality for two convex domains where one contains the other \(\Omega_2 \subset \Omega_1 \subset \mathbb{R}^n\): \[\frac{\mid\partial\Omega_1\mid}{\mid\Omega_1\mid} \frac{\mid \Omega_2\mid}{\mid \partial\Omega_2 \mid} \le n.\] Reviewer: James Adedayo Oguntuase (Abeokuta) Cited in 1 ReviewCited in 1 Document MSC: 26D15 Inequalities for sums, series and integrals 26B25 Convexity of real functions of several variables, generalizations 28A75 Length, area, volume, other geometric measure theory 31A05 Harmonic, subharmonic, superharmonic functions in two dimensions 31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions 35B50 Maximum principles in context of PDEs Keywords:Hermite-Hadamard inequality; subharmonic functions; convexity PDFBibTeX XMLCite \textit{T. 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