M. Lewandowski et al., ANDERSON INEQUALITY IS STRICT FOR GAUSSIAN AND STABLE-MEASURES, Proceedings of the American Mathematical Society, 123(12), 1995, pp. 3875-3880
Let mu be a symmetric Gaussian measure bn a separable Banach space (E,
parallel to .parallel to). Denote U = {x : parallel to x parallel to
< 1}. Then for every x epsilon supp mu, x not equal 0, the function t
--> mu(U + tx) is strictly decreasing for t epsilon (0, infinity). The
same property holds for symmetric p-stable measures on E. Using this
property we answer a question of W. Linde: if integral(U+z) x d mu(x)
= 0, then z = 0.