Anderson's theorem

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Short description: On when a function on convex body K does not decrease if K is translated inwards

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

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

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

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,

[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 → Rn is an Rn-valued random variable with probability density function f : Rn → [0, +∞) and that Y : Ω → 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

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

for any origin-symmetric convex body K ⊆ Rn.

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.