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
• 
  

  

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

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,

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

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.