Anderson's Theorem

Anderson's theorem is a result in real analysis and geometry stating that for an integrable, symmetric, unimodal, non-negative function f on n-dimensional space, the integral of f over a convex body K does not decrease when K is translated toward the origin. The theorem formalizes the intuitive picture of f as a single-peaked hill centered at the origin, though for two or more dimensions the proof is not immediate, since some points of K can carry a larger value of f than their inward translate does.