Anderson's Theorem - Statement of The Theorem

Statement of The Theorem

Let K be a convex body in n-dimensional Euclidean space Rn that is symmetric with respect to reflection in the origin, i.e. K = −K. Let f : Rn → R be a non-negative, symmetric, globally integrable function; i.e.

• f(x) ≥ 0 for all x ∈ Rn;
• f(x) = f(−x) for all x ∈ Rn;

Suppose also that the super-level sets L(f, t) of f, defined by

are convex subsets of Rn for every t ≥ 0. (This property is sometimes referred to as being unimodal.) Then, for any 0 ≤ c ≤ 1 and y ∈ Rn,