Convex hyperspaces of probability measures and extensors in the asymptotic category

CONVEX HYPERSP A CES OF PROBABIL ITY ME ASURES AND EXTENSORS IN THE ASYMPTOTIC CA TEGOR Y DU ˇ SAN REPOV ˇ S AND MYKHA ILO ZARICHNYI Abstract. The ob jects of the Dranis hnik ov asymptotic category are prop er metric spaces and the mor phi s ms are asymptotica lly Lipschitz maps. In this paper we provide an exam- ple of an asymptotically zero-dim ensional space (in the sense of Gromo v) whose space of compact con v ex subsets of probabilit y measures is not an absolute extensor in the asymp- totic category in the sense of Dranishniko v. 1. I ntroduction The notion of absolute extensor pla ys an impor tant role in different br a nches of mathemat- ics. In as y mptotic top ology , the absolute extensor s are used in constructing the ho motopy theory and the asymptotic dimension theory . Among the t wo categories widely use d in a s - ymptotic category , the Dr anishnikov and the R o e categor ies (see the definition b elow), it turns out that it is the Dranishniko v categor y (the category of prop er metric spaces and the asymptotically Lips ch itz maps) in which a richer extensor theor y can b e developed. It was prov ed in [1 0] that in gener a l, the space o f probability measures of a metric space is not a n absolute extensor for the Dra nishniko v category . This provided a negative answer to a question formulated b y Dranishniko v [2, P roblem 12 ], in connection with ex istence o f the homo topy extens io n theorem in this ca teg ory . This leads to an op en pro blem of sea r ch- ing functorial constructions that preserve the class of a bsolute extensors in the as ymptotic categorie s. In the pr esent pap er w e deal with the hyperspa ces o f compact co nv ex subsets of pr obability measures. Note that these hyper spaces play a n impo r tant ro le in the decision theory , math- ematical economics and finance, in pa rticular, in the maximum (maxmin) exp ected utility theory (cf. e.g. [3]). In the case of compact metric spa ces a s well a s in the case of compact spaces o f weight ω 1 , the h yp erspac e s of compact co nvex subsets of proba bility measures are known to b e abso- lute extensor s [1]. How ever, the extension prop erties of thes e h yp erspac es in the as ymptotic category remained unknown. Our a im is to demonstra te tha t the exa mple presented in [10] also works for the hyp e rspaces compact conv ex s ubsets o f pr obability measures. Thu s the main r esult of this pap er is that the space s mentioned a bove a re not in g e neral, a symptotic extensors in the as y mptotic categ o ry . Date : Nov ember 20, 2018. 2010 Mathematics Subje ct Classific ation. Primary: 46E27, 46E30; Secondary: 54C55, 54E35. Key wor ds and phr ases. Compact con v ex set, probability measure, asympto tically zero-dimensional space, absolute extensor. 1 2 D. REPO V ˇ S AND M. ZARICHNYI 2. Preliminaries 2.1. Asymptotic category . T ogether with Ro e’s ca tegory of prop er metric s pa ces a nd coarse maps [8], the as y mptotic category A in tro duced by Dranishnikov [2] turned out to be a n imp ortant universe for developing asymptotic to po logy . A typical metric will b e denoted by d . A map f : X → Y betw een metr ic 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 asymptotic al ly Lipschitz if it is ( λ, ε )-Lipschitz for some λ, ε > 0. The (1 , 0)-Lipsc hitz maps are als o called short . The set of all sho r t functions on a metric space X is de no ted by LIP( X ). A metric space is pr op er if every closed ball in it is compact. A map of metric spaces is (metrically) pr op er if the preimages o f the b ounded sets a re b ounded. The ob jects of the category A are the pro per metric spa ces and the morphisms ar e the pr op er asymptotically Lipschitz maps. A metric space Y (not necessar ily an ob ject of A ) is a n absolute extensor (AE) for the category A if for every pro p er a symptotically Lipschitz ma p f : A → Y defined on a c lo sed subset o f a pr op er metric space X there exists a prop er asymptotically Lipschitz extensio n ¯ f : X → Y of f . 2.2. Asymptotic dimens ion. The notio n o f a symptotic dimension was introduced by Gro- mov [4]. Let X b e a metr ic spac e . A family C of subsets of X is said to b e uniformly b ounde d if there exists M > 0 such that dia m A ≤ M for e very A ∈ C . Given D > 0, we say that a family C of subsets of X is D -disjoint if inf { d ( a, a ′ ) | a ∈ A, a ′ ∈ A ′ } > D for every A, A ′ ∈ C , A 6 = A ′ . W e say that the asymptotic dimension of X is ≤ n (written asdim X ≤ n ) if for every D > 0 there exists a cover U of X s uch that U = U 0 ∪ · · · ∪ U n , where every family U i is D -discrete. If we requir e in the definition of the absolute extenso r that asdim X ≤ n , then the definition of the absolute extensor in asymptotic dimension n (briefly AE ( n )) is o bta ined. It is easy to see that for a prop er metric space X , the inequality asdim X ≤ 0 is equiv a lent to the c ondition that for every C > 0 the dia meters 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 b ounded from ab ov e. 2.3. Con v ex h yp erspaces of probabil it y measures. Let P ( X ) denote the space o f pro b- ability mea sures of compact supp or ts on a metriza ble space X . F or a ny x ∈ X , we denote the Dirac measure concentrated at x b y δ x . If d is a metric o n X , we deno te b y ˆ d the Ka n torovich metric ge nerated by d , ˆ d ( µ, ν ) = sup      Z ϕdµ − Z ϕdν     | ϕ ∈ LIP( X )  (cf. e.g . [5]). By cc P ( X ) we denote the set of all nonempty co mpa ct conv ex subsets in P ( X ); as usual, a subse t A ⊂ P ( X ) is co n vex if tµ + (1 − t ) ν ∈ A , for all µ, ν ∈ P ( X ) and t ∈ [0 , 1]. The s e t cc P ( X ) is endow ed with the Hausdorff metric, which w e shall deno te by ˆ d H : ˆ d H ( A, B ) = inf { r > 0 | A ⊂ O r ( B ) , B ⊂ O r ( A ) } (here O t ( Y ) sta nds for the t -neighborho o d of Y ⊂ P ( X )). Note tha t, clea rly , the map x 7→ { δ x } : X → cc P ( X ) is an iso metric embedding. Given a map f : X → Y of metric spaces, we de fine the map P ( f ) : P ( X ) → P ( Y ) a s follows: R ϕdP ( f )( µ ) = R ϕf dµ . The map cc P ( f ) : cc P ( X ) → cc P ( Y ) is then defined b y the HYPERSP A CES OF PROBABILITY MEASURES AND EXTENSORS 3 formula: cc P ( f )( A ) = { P ( f )( µ ) | µ ∈ A } . It c a n b e easily seen that the ma p cc P ( f ) is shor t if such is f . Let b : P ( R n ) → R n denote the bary cent er map. Recall tha t this map assig ns to every µ ∈ P ( R n ) the unique p o int b ( µ ) with the pr op erty that L ( b ( µ )) = R Ldµ , for e very contin uous linear functional L on R n . Since b is known to b e contin uous and linear, the image b ( A ) of every A ∈ cc P ( R n ) is a compa ct co nv ex subset of R n , i.e., a n element of the space cc( R n ) o f compact co nv e x subsets in R n endow ed with the Hausdorff metric. Let p : cc( R n ) → R n denote the map defined by the condition: y = π ( A ) ⇔ y ∈ A a nd k y k = inf {k z k | z ∈ A } . The pro of o f the following statement uses simple geometr ic a rguments and will therefore be o mitted. Lemma 2.1. The map π is wel l- define d and short. 3. The Example Our example describ ed b elow is a modifica tion of the second author’s example of a pr op er metric space who se space of pro bability meas ur es is not an AE (even AE (0)) in the asymp- totic category [11]. F or the sake of completeness we shall provide her e the details of the construction. F or e very n , the Euclidean spa c e R n can natura lly b e identified with the subspace { ( x i ) | x i = 0 for all j > n } of the space ℓ 2 . W e endow the subspace X ′ = S n ∈ N { n 2 } × R n ⊂ R × ℓ 2 with the metric d (( m, ( x i )) , ( n, ( y i ))) = ( | m − n | 2 + k ( x i ) − ( y i ) k 2 ) 1 / 2 . Obviously , X is a pro p er metric space. F or every n we denote b y p n : X ′ → R n a map defined by the for m ula p n ( m, ( x i )) = ( x 1 , . . . , x n ). Clearly , p n is a sho rt map. It was shown in [6] (cf. Theor em 1.5 therein) that for any n ≥ 2 there exists a metric space X n which co nt ains the Euclidean space R n as a metric s ubspace and such that there is no ( λ, ε )-Lipschitz retractio n from X n onto R n with λ < n 1 / 4 . In the sequel we shall need an explicit cons truction of these spaces. F ollowing [6], for every natural k and natur al n ≥ 2 we define graphs G n,k as follows: the se t o f vertices V ( G n,k ) is the union of I ( G n,k ) and T ( G n,k ), where I ( G n,k ) = { x = ( x 1 , . . . , x n ) ∈ R n | | x i | = k for all i } , T ( G n,k ) = { x = ( x 1 , . . . , x n ) ∈ R n | | x i | = 2 k for a ll i } ; the set o f edges E ( G n,k ) is defined by the co nditio n: { x, y } ∈ E ( G n,k ) if and only if x, y ∈ V ( G n,k ) and either k x − y k = 2 k or y = 2 x (we supp os e that the spaces R n are endow ed with the E uclidean metr ic ). The set V ( G n,k ) is equipp ed with the metric d = d n,k , d ( x, y ) = inf ( l X i =1 k x i − 1 − x i k ∞ | ( x = x 0 , x 1 , . . . , x l = y ) is a path in G n,k ) (as usua l, k x k ∞ denotes the max- norm of x ∈ R n .) Define s paces X and Y a s follows: X = ∞ [ n =2 ∞ [ k = n { n 2 } × T ( G n 2 ,k 2 ) , Y = ∞ [ n =2 ∞ [ k = n { n 2 } × V ( G n 2 ,k 2 ) 4 D. REPO V ˇ S AND M. ZARICHNYI where the metric on X is inher ited from X ′ and the metric on Y is the maximal metric that agrees with the alr eady defined metric on X and the metric d n,k on every V ( G n,k ). It easily follows fro m the construction that X and Y are prop er metric spaces, i.e. ob jects o f the category A . W e a re go ing to s how that a sdim Y = 0 (and consequently asdim X = 0). Let C > 0 and suppo se that y 1 , . . . , y m is a C - chain in Y . Deno te by k the minimal natural num b e r such that C < ( k + 1) 2 . If { y 1 , . . . , y n } ⊂ k [ j =2 k [ l = j { j 2 } × V ( G j 2 ,l 2 ) then dia m { y 1 , . . . , y n } ≤ p ( k 2 ) + (3 k 2 ) 2 ≤ √ 10 C . Otherwise { y 1 , . . . , y n } ∩ Y \ k [ j =2 k [ l = j { j 2 } × V ( G j 2 ,l 2 ) 6 = ∅ and { y 1 , . . . , y n } is a singleto n. It was pr ov ed in [6] that the following holds for the spaces X n 2 = R n 2 ∪ ∞ [ k = n { n 2 } × V ( G n 2 ,k 2 ) endow ed with the maximal metric which agre e s with the initial metric o n R n 2 and the metric on S ∞ k = n { n 2 } × V ( G n 2 ,k 2 ) inherited fro m Y (note that these tw o metrics co incide o n the int ersec tio n of their domains): ther e is no ( λ, ε )-retra ction of X n 2 to R n 2 with λ < √ n . Now, let f : X → cc P ( X ) b e the map that sends x ∈ X to { δ x } ∈ cc P ( X ). The map f is an isometric embedding and we ar e g oing to show that there is no asy mptotically Lipschitz extension o f f ont o the whole space Y . Ass ume the contrary and let ¯ f : Y → c c P ( X ) b e such an extensio n. W e regar d ¯ f as a map into F ( X ′ ) ⊃ cc P ( X ). Then there exist λ > 0 and ε > 0 such tha t d ( ¯ f ( x ) , ¯ f ( x ′ )) ≤ λd ( x, x ′ ) + ε for a ll x, x ′ ∈ Y . Let n > λ 2 . Since the ma ps cc P ( p n 2 ), b : P ( R n 2 ) → R n 2 and π ar e shor t, we conclude that the ma p x 7→ π ( { b ( µ ) | µ ∈ cc P ( p n 2 )( ¯ f ( x )) } ) : X n 2 → R n 2 is a ( λ, ε )-Lipschitz retra ction from X n 2 onto R n 2 , which c o nt radic ts to the choice of λ . This demonstrates that the space cc P ( X ) is not an AE(0) for the asymptotic ca tegory A . 4. Epilogue W e conjecture that the spaces of capacities (non-additive mea sures; cf. e .g . [13, 14]) are alwa ys absolute extensors in the as ymptotic catego ry A . Note that the scheme o f our pro of of the ma in re s ult of this pap er do es not work for non-additive s itua tion, b ecause o ne do es not have the “ba rycenter map” in this case. Ackno wledgements This research w as supp or ted by the Slov enian Resear ch Agency grants P1-02 92-01 01, J1- 9643- 0101 a nd J1-2 057-0 101. W e tha nk the r eferee for comments and suggestions . HYPERSP A CES OF PROBABILITY MEASURES AND EXTENSORS 5 References [1] L. Bazylevych , D. Rep ov ˇ s, M. Zarichn yi, Sp ac es of idemp otent me asur es of c omp act metric sp ac e s , T op ol. Appl. 157 (2010), 136-144. [2] A. Dranishniko v, Asymptotic top olo gy , Russian Math. Surv. 55 (2000), 71–116. [3] I. Gilb oa, D. Schme idler, Maxmin exp e cte d utility with a non-unique prior , J. M ath. Economics 18 (1989), 141–53. [4] M. Gromov, Asymptotic i nv ariants of infinite gr oups , Geometric Group Theory , V ol. 2 (Sussex, 1991), London Math. Soc. Lecture Note Ser. V ol. 182, Cam bridge Univ. Press, Cambridge, 1993, pp. 1–295. [5] J.E. Hutchinson , F r actals and self-similarity , Indiana Univ. Math. J. 30 (1981), 713–747. [6] U. 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