By Silvio Levy

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1−δ 1−δ Proof. Clearly the value of M plays no real role here so assume it is 1. We start with the upper bound. Let C be the maximum possible ratio x /|x| for nonzero x and let θ be a point of S k−1 with θ = C. Choose ψ in the δ-net with |θ − ψ| ≤ δ. Then θ − ψ ≤ C|θ − ψ| ≤ Cδ, so C = θ ≤ ψ + θ − ψ ≤ (1 + γ) + Cδ. Hence (1 + γ) . 1−δ To get the lower bound, pick some θ in the sphere and some ψ in the δ-net with |ψ − θ| ≤ δ. Then C≤ (1 − γ) ≤ ψ ≤ θ + ψ − θ ≤ θ + (1 + γ)δ (1 + γ) |ψ − θ| ≤ θ + . 1−δ 1−δ Hence θ ≥ 1−γ− δ(1 + γ) 1−δ = (1 − γ − 2δ) .

T, y) (s, x) Ar As At Figure 21. The section As contains the weighted average of Ar and At . ) Brunn’s Theorem says that the volumes of the three sets Ar , As , and At in Rn−1 satisfy 1/(n−1) vol (As ) 1/(n−1) ≥ (1 − λ) vol (Ar ) 1/(n−1) + λ vol (At ) . The Brunn–Minkowski inequality makes explicit the fact that all we really know about As is that it includes the Minkowski combination of Ar and At . Since we have now eliminated the role of the ambient space Rn , it is natural to rewrite the inequality with n in place of n − 1.

To begin with, a probability measure µ on a set Ω is just a measure of total mass µ(Ω) = 1. Real-valued functions on Ω are called random variables and the integral of such a function X : Ω → R, its mean, is written EX and called the expectation of X. The variance of X is E(X − EX)2 . It is customary to suppress the reference to Ω when writing the measures of sets defined by random variables. Thus µ({ω ∈ Ω : X(ω) < 1}) is written µ(X < 1): the probability that X is less than 1. Two crucial, and closely related, ideas distinguish probability theory from general measure theory.