Triple product property

29 yr old Orthopaedic Surgeon Grippo from Saint-Paul, spends time with interests including model railways, top property developers in singapore developers in singapore and dolls. Finished a cruise ship experience that included passing by Runic Stones and Church. 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;
• $\underset{{\mathbb{ℝ}}^n}{\int }f(x)\mathrm{d}x<+\mathrm{\infty }.$

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

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

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,

$\underset{K}{\int }f(x+cy)\mathrm{d}x\ge \underset{K}{\int }f(x+y)\mathrm{d}x.$

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

$\mathrm{Pr}(X\in K)\ge \mathrm{Pr}(X+Y\in K)$

for any origin-symmetric convex body K ⊆ Rⁿ.

References

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