Anderson's theorem
# 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(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.

## Statement of the theorem

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

* f(x) ≥ 0 for all x ∈ $R^n;$
* f(x) = f(−x) for all x ∈ $R^n;$
* $\int _{\mathbb {R} ^{n}}f(x)\,\mathrm {d} x<+\infty .$

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 $R^n$ for every t ≥ 0. (This property is sometimes referred to as being unimodal.) Then, for any 0 ≤ c ≤ 1 and y ∈ $R^n,$

    $\int _{K}f(x+cy)\,\mathrm {d} x\geq \int _{K}f(x+y)\,\mathrm {d} x.$

## Application to probability theory

Given a probability space (Ω, Σ, Pr), suppose that X : Ω → $R^n$ is an $R^n-valued$ random variable with probability density function f : $R^n$ → [0, +∞) and that Y : Ω → $R^n$ 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 ⊆ $R^n.$

