# Anderson's theorem

En mathématiques, 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; toutefois, pour 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, voir Anderson orthogonality theorem. For the superconductivity theorem, voir Anderson's theorem (superconductivity).

Dans mathématiques, Anderson's théorème is a result in real analysis et géométrie which says that the integral of an integrable, symmetric, unimodal, non-negative function F over an n-dimensionnel convex body K does not decrease if K is translated inwards towards the origin. This is a natural statement, since the graph de F can be thought of as a hill with a single peak over the origin; toutefois, pour 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.

## Énoncé du théorème[]

Laisser K be a convex body in n-dimensionnel Espace euclidien Rn C'est symmetric with respect to reflection in the origin, c'est à dire. K = −K. Laisser F : Rn →R be a non-négatif, symmetric, globally integrable function; c'est à dire.

• F(X≥ 0 for all X Rn
• F(X) =F(−X) pour tous X Rn

• $"{displaystyle$

Suppose also that the super-level sets L(Ft) de F, Défini par

$"{displaystyle$geq t},}

sommes convex subsets de Rn pour chaque t ≥ 0. (This property is sometimes referred to as being unimodal.) Alors, for any 0 ≤c ≤ 1 and y Rn,

$"{displaystyle$x+cy),mathrm {ré} xgeq int _{K}F(x+y),mathrm {ré} X.}

## Application to probability theory[]

Given a probability space (Oh, S, Pr), supposer que X : Ω →Rn est un Rn-estimé random variable avec probability density function F : Rn →[0, +∞) et cela Oui : Ω →Rn est un independent random variable. The probability density functions of many well-known probability distributions are p-concave pour certains p, and hence unimodal. If they are also symmetric (par exemple. la Laplace et normal distributions), then Anderson's theorem applies, dans quel cas

$"{displaystyle$Xin K)geq Pr(X+Yin K)}

for any origin-symmetric convex body K Rn.

## Références[]

• Gardner, Richard J.. (2002). "The Brunn-Minkowski inequality". Taureau. Amer. Math. Soc. (N.S.). 39 (3): 355–405 (electronic). est ce que je:10.1090/S0273-0979-02-00941-2.

Si vous voulez connaître d'autres articles similaires à Anderson's theorem vous pouvez visiter la catégorie Théorèmes de probabilité.

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