Anderson's theorem
In mathematics, Anderson's theorem is a result in real analysis and geometry which says that the integral of an integrable, symmetric, unimodal, non-negative function f over an n-dimensional convex body K does not decrease if K is translated inwards towards the origin. This is a natural statement, since the graph of f can be thought of as a hill with a single peak over the origin; however, for n ≥ 2, the proof is not entirely obvious, as there may be points x of the body K where the value f is larger than at the corresponding translate of x.
Anderson's theorem also has an interesting application to probability theory.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 ≥ 0 for all x ∈ Rn;
- f = f for all x ∈ Rn;
-
Suppose also that the super-level sets L of f, defined by
are convex subsets of Rn for every t ≥ 0. Then, for any 0 ≤ c ≤ 1 and y ∈ Rn,Application to probability theory
Given a probability space and that Y : Ω → Rn is an independent random variable. The probability density functions of many well-known probability distributions are p-concave for some p, and hence unimodal. If they are also symmetric, then Anderson's theorem applies, in which case
for any origin-symmetric convex body K ⊆ Rn.
