ANDERSON INEQUALITY IS STRICT FOR GAUSSIAN AND STABLE-MEASURES

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.