Anderson's theorem

Na matemática, 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; Contudo, 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

This article is about Anderson's theorem in mathematics. For the Anderson orthogonality theorem in physics, Vejo Anderson orthogonality theorem. For the superconductivity theorem, Vejo Anderson's theorem (superconductivity).

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

• $"{displaystyle$

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

$"{displaystyle$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,

$"{displaystyle$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

$"{displaystyle$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.

Se você quiser conhecer outros artigos semelhantes a Anderson's theorem você pode visitar a categoria teoremas de probabilidade.

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