Restricted mean value property on \(\mathbb{R}^ d, d\leq 2\)
Let \(r\) be a strictly positive real function on \(\mathbb{R}\) such that \(r(x) \leq |x |+ c\) where \(c\) is some positive constant. A nonnegative lower-semi-continuous function \(f\) is said to be \(r\) supermedian if the integral of \(f\) on the intervals of center \(x\) and radius \(r(x)\) is not greater than \(f(x)\) for any \(x\). It is proved in this (3 page) paper that such a function is necessarily constant using Bauer's minimum principle.
scientific article; zbMATH DE number 783958