Dentro matemática, Anderson's teorema is a result in real analysis e geometria 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 do f can be thought of as a hill with a single peak over the origin; Contudo, por 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 teoria da probabilidade.

Declaração do teorema[]

Deixar K be a convex body in n-dimensional espaço euclidiano Rⁿ isso é symmetric with respect to reflection in the origin, ou seja. K = −K. Deixar f : Rⁿ →R be a non-negativo, symmetric, globally integrable function; ou seja.

• f(x) ≥ 0 for all x ∈Rⁿ
• f(x) =f(−x) para todos x ∈Rⁿ
• 
  

  

Suppose also that the super-level sets eu(f, t) do f, definido por

xin mathbb {R} ^{n}|f(x)geq t},}

são convex subsets do Rⁿ para cada t ≥ 0. (This property is sometimes referred to as being unimodal.) Então, for any 0 ≤c ≤ 1 and y ∈Rⁿ,

x+cy),matemática {d} xgeq int _{K}f(x+y),matemática {d} x.}

Application to probability theory[]

Given a probability space (Oh, S, Pr), Suponha que X : Ω →Rⁿ é um Rⁿ-valorizado random variable com probability density function f : Rⁿ →[0, +∞) and that S : Ω →Rⁿ é um independent random variable. The probability density functions of many well-known probability distributions are p-concave para alguns p, and hence unimodal. If they are also symmetric (por exemplo. a Laplace e normal distributions), then Anderson's theorem applies, nesse caso

Xin K)geq Pr(X+Yin K)}

for any origin-symmetric convex body K ⊆Rⁿ.

Referências[]

• Gardner, Richard J. (2002). "The Brunn-Minkowski inequality". Touro. América. Matemática. Soc. (N.S.). 39 (3): 355-405 (electronic). doi:10.1090/S0273-0979-02-00941-2.