# Anderson's theorem

In Mathematik, 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; jedoch, 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

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

• $"{displaystyle$ Suppose also that the super-level sets L(ft) von f, definiert von

$"{displaystyle$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,

$"{displaystyle$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

$"{displaystyle$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.

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