# Anderson's theorem

In mathematics, 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; however, 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 also has an interesting application to probability theory.

Statement of the theorem Let K be a convex body in n-dimensional Euclidean space Rn that is symmetric with respect to reflection in the origin, i.e. K = −K. Let f : Rn → R be a non-negative, symmetric, globally integrable function; i.e.

f(x) ≥ 0 for all x ∈ Rn; f(x) = f(−x) for all x ∈ Rn; {displaystyle int _{mathbb {R} ^{n}}f(x),mathrm {d} x<+infty .} Suppose also that the super-level sets L(f, t) of f, defined by {displaystyle L(f,t)={xin mathbb {R} ^{n}|f(x)geq t},} are convex subsets of Rn for every t ≥ 0. (This property is sometimes referred to as being unimodal.) Then, for any 0 ≤ c ≤ 1 and y ∈ Rn, {displaystyle int _{K}f(x+cy),mathrm {d} xgeq int _{K}f(x+y),mathrm {d} x.} Application to probability theory Given a probability space (Ω, Σ, Pr), suppose that X : Ω → Rn is an Rn-valued random variable with probability density function f : Rn → [0, +∞) and that Y : Ω → Rn is an independent random variable. The probability density functions of many well-known probability distributions are p-concave for some p, and hence unimodal. If they are also symmetric (e.g. the Laplace and normal distributions), then Anderson's theorem applies, in which case {displaystyle Pr(Xin K)geq Pr(X+Yin K)} for any origin-symmetric convex body K ⊆ Rn. References Gardner, Richard J. (2002). "The Brunn-Minkowski inequality". Bull. Amer. Math. Soc. (N.S.). 39 (3): 355–405 (electronic). doi:10.1090/S0273-0979-02-00941-2. Categories: Theorems in geometryProbability theoremsTheorems in real analysis

Si quieres conocer otros artículos parecidos a Anderson's theorem puedes visitar la categoría Probability theorems.

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