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(x) is larger than at the corresponding translate of x. Anderson's theorem, named after Theodore Wilbur Anderson, 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.
Suppose also that the super-level sets L(f, t) of f, defined by are convex subsets of Rn for every t ≥ 0.
If they are also symmetric (e.g. the Laplace and normal distributions), then Anderson's theorem applies, in which case for any origin-symmetric convex body K ⊆ Rn.