In matematica, 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-dimensionale convex body K does not decrease if K is translated inwards towards the origin. This is a natural statement, since the graph di f can be thought of as a hill with a single peak over the origin; però, per 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.

Enunciato del teorema[]

Permettere K be a convex body in n-dimensionale Euclidean space Rⁿ questo è symmetric with respect to reflection in the origin, cioè. K = −K. Permettere f : Rⁿ →R be a non-negativo, symmetric, globally integrable function; cioè.

• f(X) ≥ 0 for all X ∈Rⁿ
• f(X) =f(−X) per tutti X ∈Rⁿ
• 
  

  

Suppose also that the super-level sets l(f, t) di f, definito da

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

sono convex subsets di Rⁿ per ogni t ≥ 0. (This property is sometimes referred to as being unimodal.) Quindi, for any 0 ≤c ≤ 1 and y ∈Rⁿ,

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

Application to probability theory[]

Given a probability space (Oh, S, pr), supporre che X : Ω →Rⁿ è un Rⁿ-apprezzato random variable insieme a probability density function f : Rⁿ →[0, +∞) e quello Y : Ω →Rⁿ è un independent random variable. The probability density functions of many well-known probability distributions are p-concave per alcuni p, and hence unimodal. If they are also symmetric (per esempio. il Laplace e normal distributions), then Anderson's theorem applies, in quale caso

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

for any origin-symmetric convex body K ⊆Rⁿ.

Riferimenti[]

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