The objects of the Dranishnikov asymptotic category are proper metric spaces and the morphisms are asymptotically Lipschitz maps. In this paper we provide an example of an asymptotically zero-dimensional space (in the sense of Gromov) whose space of compact convex subsets of probability measures is not an absolute extensor in the asymptotic category in the sense of Dranishnikov.
The notion of absolute extensor plays an important role in different branches of mathematics. In asymptotic topology, the absolute extensors are used in constructing the homotopy theory and the asymptotic dimension theory. Among the two categories widely used in asymptotic category, the Dranishnikov and the Roe categories (see the definition below), it turns out that it is the Dranishnikov category (the category of proper metric spaces and the asymptotically Lipschitz maps) in which a richer extensor theory can be developed.
It was proved in [10] that in general, the space of probability measures of a metric space is not an absolute extensor for the Dranishnikov category. This provided a negative answer to a question formulated by Dranishnikov [2,Problem 12], in connection with existence of the homotopy extension theorem in this category. This leads to an open problem of searching functorial constructions that preserve the class of absolute extensors in the asymptotic categories.
In the present paper we deal with the hyperspaces of compact convex subsets of probability measures. Note that these hyperspaces play an important role in the decision theory, mathematical economics and finance, in particular, in the maximum (maxmin) expected utility theory (cf. e.g. [3]).
In the case of compact metric spaces as well as in the case of compact spaces of weight ω 1 , the hyperspaces of compact convex subsets of probability measures are known to be absolute extensors [1]. However, the extension properties of these hyperspaces in the asymptotic category remained unknown. Our aim is to demonstrate that the example presented in [10] also works for the hyperspaces compact convex subsets of probability measures. Thus the main result of this paper is that the spaces mentioned above are not in general, asymptotic extensors in the asymptotic category.
2.1. Asymptotic category. Together with Roe’s category of proper metric spaces and coarse maps [8], the asymptotic category A introduced by Dranishnikov [2] turned out to be an important universe for developing asymptotic topology.
A typical metric will be denoted by d. A map f : X → Y between metric spaces is called (λ, ε)-Lipschitz for λ > 0, ε ≥ 0 if d(f (x), f (x ′ )) ≤ λd(x, x ′ ) + ε for every x, x ′ ∈ X. A map is called asymptotically Lipschitz if it is (λ, ε)-Lipschitz for some λ, ε > 0. The (1, 0)-Lipschitz maps are also called short. The set of all short functions on a metric space X is denoted by LIP(X).
A metric space is proper if every closed ball in it is compact. A map of metric spaces is (metrically) proper if the preimages of the bounded sets are bounded. The objects of the category A are the proper metric spaces and the morphisms are the proper asymptotically Lipschitz maps.
A metric space Y (not necessarily an object of A) is an absolute extensor (AE) for the category A if for every proper asymptotically Lipschitz map f : A → Y defined on a closed subset of a proper metric space X there exists a proper asymptotically Lipschitz extension f : X → Y of f . 2.2. Asymptotic dimension. The notion of asymptotic dimension was introduced by Gromov [4]. Let X be a metric space. A family C of subsets of X is said to be uniformly bounded if there exists
We say that the asymptotic dimension of X is ≤ n (written asdimX ≤ n) if for every D > 0 there exists a cover U of X such that U = U 0 ∪ • • • ∪ U n , where every family U i is D-discrete. If we require in the definition of the absolute extensor that asdimX ≤ n, then the definition of the absolute extensor in asymptotic dimension n (briefly AE(n)) is obtained.
It is easy to see that for a proper metric space X, the inequality asdimX ≤ 0 is equivalent to the condition that for every C > 0 the diameters of the C-chains in X (i.e. the sequences x 1 , x 2 , . . . , x k with d(x i , x i+1 ) ≤ C for every i = 1, 2, . . . , k -1) are bounded from above.
2.3. Convex hyperspaces of probability measures. Let P (X) denote the space of probability measures of compact supports on a metrizable space X. For any x ∈ X, we denote the Dirac measure concentrated at x by δ x . If d is a metric on X, we denote by d the Kantorovich metric generated by d,
(cf. e.g. [5]). By ccP (X) we denote the set of all nonempty compact convex subsets in P (X); as usual, a subset A ⊂ P (X) is convex if tµ + (1t)ν ∈ A, for all µ, ν ∈ P (X) and t ∈ [0, 1]. The set ccP (X) is endowed with the Hausdorff metric, which we shall denote by dH :
(here O t (Y ) stands for the t-neighborhood of Y ⊂ P (X)). Note that, clearly, the map x → {δ x } : X → ccP (X) is an isometric embedding.
Given a map f : X → Y of metric spaces, we define the map P (f ) : P (X) → P (Y ) as follows: ϕdP (f )(µ) = ϕf dµ. The map ccP (f ) : ccP (X) → ccP (Y ) is then defined by the formula: ccP (f )(A) = {P (f )(µ) | µ ∈ A}. It can be easily seen that the map ccP (f ) is short if such is f . Let b : P (R n ) → R n denote the barycenter map. Recall that this map assigns to every µ ∈
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