Characterization of Gaussian measures by the isoperimetric property of half-spaces
Let \(\mu\) be a probability measure on the real line \(R\), and let \(\mu^n\) be its \(n\)th power on \(R^n\). Let \(A^h\) be the open \(h\)-neighborhood, \(h>0\), of the subset \(A\subset R^n\) in the sense of Euclidean distance. The authors prove that for \(n\geq 2\), if for all \(p\in (0,1)\) and all \(h>0\) the minimal value of \(\mu^n(A^h)\) is attained at a standard half-space, then \(\mu\) is Gaussian, possibly degenerate. The hypotheses of this result can be weakened if in addition \(\mu\) is symmetric and has a finite second moment.
scientific article; zbMATH DE number 1300647