# Anderson's theorem

__: On when a function on convex body K does not decrease if K is translated inwards__

**Short description**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 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};- [math]\displaystyle{ \int_{\mathbb{R}^{n}} f(x) \, \mathrm{d} x \lt + \infty. }[/math]

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

- [math]\displaystyle{ L(f, t) = \{ x \in \mathbb{R}^{n} | f(x) \geq t \}, }[/math]

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},

- [math]\displaystyle{ \int_{K} f(x + c y) \, \mathrm{d} x \geq \int_{K} f(x + y) \, \mathrm{d} x. }[/math]

## 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

- [math]\displaystyle{ \Pr ( X \in K ) \geq \Pr ( X + Y \in K ) }[/math]

for any origin-symmetric convex body *K* ⊆ **R**^{n}.

## References

- Gardner, Richard J. (2002). "The Brunn-Minkowski inequality".
*Bull. Amer. Math. Soc. (N.S.)***39**(3): 355–405 (electronic). doi:10.1090/S0273-0979-02-00941-2.

Original source: https://en.wikipedia.org/wiki/Anderson's theorem.
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