xvi Notation Whenever Ω is a smooth subset of Rn, the group of all diffeomorphisms s : Ω → Ω with det(∇s) ≡ 1 will be denoted by G(Ω). We shall refer to a measurable set X ⊂ Rn as a small set if it has Hausdorff dimension at most n − 1. The vector space of real n × n matrices will be denoted by Mn(R). The trace of a matrix M will be denoted by tr M. The n × n identity matrix will be denoted by In. Whenever M is an element of Mn(R), its transposed matrix will be denoted by M T thus M T = (mij) with mij = mji. The sets of symmetric matrices (M T = M), symmetric matrices with nonnegative eigenvalues (M ≥ 0), antisymmetric matrices (M T = −M) and orthogonal matrices (MM T = In) will be respectively denoted by Sn(R), Sn +(R), An(R) and On(R). Finally, let us say a word about where to find the definitions of the basic objects in optimal mass transportation: the notations I[π], Π(μ, ν), J(ϕ, ψ), Φc are defined in Theorem 1.3 of Chapter 1 Tc(μ, ν) in formula (5) Wp(μ, ν) and Tp(μ, ν) in Theorem 7.3 of Chapter 7.

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