 By Claude Flament

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4) dq is 2k-gonal distance. Every CAT(0) space (cf. Chap. 6) has roundness 2, but some of them have generalized roundness 0 (Lafont and Prassidis 2006). • Type of metric space The Enﬂo type of a metric space (X, d) is p if there exists a constant 1 ≤ C < ∞ such that, for every n ∈ N and every function n p p f : {−1, 1}n → X, j=1 ∈{−1,1}n d (f ( ), f (− )) is at most C p j+1 , ∈{−1,1}n d (f ( 1 , . . , j−1 , j , j+1 , . . , n ), f ( 1 , . . , j−1 , − j , . . , n )). , for every x1 , . . , xn ∈ V , n || ∈{−1,1}n n p p j xj || ≤ C j=1 ||xj ||p .

Also, hypdim(X, d) ≤ asdim(X, d) since the asymptotic dimension asdim(X, d) corresponds to the case N = 1 in the deﬁnition of hypdim(X, d). The hyperbolic dimension is preserved under a quasi-isometry. Asymptotic dimension The asymptotic dimension asdim(X, d) of a metric space (X, d) (Gromov 1993) is the smallest integer n such that, for every r > 0, there exist a constant D = D(r) and a covering of X by its subsets of diameter at most D such that every ball of radius r meets at most n + 1 elements of the covering.

Two subsets A and B of a metric space (X, d) are called (Gowers 2000) similar if there exist short mappings f : A → X, g : B → X and a small > 0 such that every point of A is within of some point of B, every point of B is within of some point of A, and |d(x, g(f (x))) − d(y, f (g(y)))| ≤ for every x ∈ A and y ∈ B. • Category of metric spaces A category Ψ consists of a class ObΨ, whose elements are called objects of the category, and a class M orΨ, elements of which are called morphisms of the category.