Im Mathematik, Anderson's Satz is a result in real analysis und Geometrie 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 von f can be thought of as a hill with a single peak over the origin; jedoch, zum 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 Wahrscheinlichkeitstheorie.

Aussage des Theorems[]

Lassen K be a convex body in n-dimensional Euclidean space Rⁿ das ist symmetric with respect to reflection in the origin, d.h. K = −K. Lassen f : Rⁿ →R be a non-Negativ, symmetric, globally integrable function; d.h.

• f(x) ≥ 0 for all x ∈Rⁿ
• f(x) =f(−x) für alle x ∈Rⁿ
• 
  

  

Suppose also that the super-level sets L(f, t) von f, definiert von

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

sind convex subsets von Rⁿ für jeden t ≥ 0. (This property is sometimes referred to as being unimodal.) Dann, for any 0 ≤c ≤ 1 and j ∈Rⁿ,

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

Application to probability theory[]

Given a probability space (Oh, S, Pr), nehme an, dass X : Ω →Rⁿ ist ein Rⁿ-geschätzt random variable mit probability density function f : Rⁿ →[0, +∞) und das Y : Ω →Rⁿ ist ein independent random variable. The probability density functions of many well-known probability distributions are p-concave für einige p, and hence unimodal. If they are also symmetric (z.B. das Laplace und normal distributions), then Anderson's theorem applies, in welchem ​​Fall

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

for any origin-symmetric convex body K ⊆Rⁿ.

Verweise[]

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