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 &ge; 2, the proof is not entirely obvious, as there may be points x of the body K where the value f(x) is larger than at the corresponding translate of x.

Anderson's theorem also has an interesting application to probability theory.

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 = &minus;K. Let f : Rn &rarr; R be a non-negative, symmetric, globally integrable function; i.e.
 * f(x) &ge; 0 for all x &isin; Rn;
 * f(x) = f(&minus;x) for all x &isin; Rn;
 * $$\int_{\mathbb{R}^{n}} f(x) \, \mathrm{d} x < + \infty.$$

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


 * $$L(f, t) = \{ x \in \mathbb{R}^{n} | f(x) \geq t \},$$

are convex subsets of Rn for every t &ge; 0. (This property is sometimes referred to as being unimodal.) Then, for any 0 &le; c &le; 1 and y &isin; Rn,


 * $$\int_{K} f(x + c y) \, \mathrm{d} x \geq \int_{K} f(x + y) \, \mathrm{d} x.$$

Application to probability theory
Given a probability space (&Omega;, &Sigma;, Pr), suppose that X : &Omega; &rarr; Rn is an Rn-valued random variable with probability density function f : Rn &rarr; [0, +&infin;) and that Y : &Omega; &rarr; 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 (e.g. the Laplace and normal distributions), then Anderson's theorem applies, in which case


 * $$\Pr ( X \in K ) \geq \Pr ( X + Y \in K )$$

for any origin-symmetric convex body K &sube; Rn.