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 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(x) ≥ 0 for all x ∈ Rn;

f(x) = f(−x) for all x ∈ Rn;

\( {\displaystyle \int _{\mathbb {R} ^{n}}f(x)\,\mathrm {d} x<+\infty .} \)

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

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

are convex subsets of Rn for every t ≥ 0. (This property is sometimes referred to as being unimodal.) Then, for any 0 ≤ c ≤ 1 and y ∈ Rn,

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

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

\( {\displaystyle \Pr(X\in K)\geq \Pr(X+Y\in K)} \)

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.

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