For positive subharmonic f on convex Omega in R^n the volume average is at most c_n times the surface average with n-1 <= c_n <= 2 n^{3/2}, plus the sharp geometric inequality |partial Omega1|/|Omega1| * |Omega2|/|partial Omega2| <= n for nested convex domains.
Bessenyei: The Hermite-Hadamard inequality on simpl ices, American Mathematical Monthly 115(4), 339–345 (2008
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Improved Bounds for Hermite-Hadamard Inequalities in Higher Dimensions
For positive subharmonic f on convex Omega in R^n the volume average is at most c_n times the surface average with n-1 <= c_n <= 2 n^{3/2}, plus the sharp geometric inequality |partial Omega1|/|Omega1| * |Omega2|/|partial Omega2| <= n for nested convex domains.