Geometric Constructions in the Class of Busemann Nonpositively Curved Spaces

Geometric Constructions in the Class of Busemann Nonpositively Curved Spaces

The procedures of gluing and the Hausdorff limit in the class of metric spaces nonpositively curved in the sense of Busemann are studied in the paper. Conditions under which the resulting spaces belong to the same class are found.

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Дата:2009
Автор: Andreev, P.D.
Формат: Стаття
Мова:English
Опубліковано: Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2009
Назва видання:Журнал математической физики, анализа, геометрии
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Назва журналу:Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:Geometric Constructions in the Class of Busemann Nonpositively Curved Spaces / P.D. Andreev // Журнал математической физики, анализа, геометрии. — 2009. — Т. 5, № 1. — С. 25-37. — Бібліогр.: 13 назв. — англ.

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spelling irk-123456789-1065312016-10-01T03:01:45Z Geometric Constructions in the Class of Busemann Nonpositively Curved Spaces Andreev, P.D. The procedures of gluing and the Hausdorff limit in the class of metric spaces nonpositively curved in the sense of Busemann are studied in the paper. Conditions under which the resulting spaces belong to the same class are found. 2009 Article Geometric Constructions in the Class of Busemann Nonpositively Curved Spaces / P.D. Andreev // Журнал математической физики, анализа, геометрии. — 2009. — Т. 5, № 1. — С. 25-37. — Бібліогр.: 13 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106531 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
description The procedures of gluing and the Hausdorff limit in the class of metric spaces nonpositively curved in the sense of Busemann are studied in the paper. Conditions under which the resulting spaces belong to the same class are found.
format Article
author Andreev, P.D.
spellingShingle Andreev, P.D.
Geometric Constructions in the Class of Busemann Nonpositively Curved Spaces
Журнал математической физики, анализа, геометрии
author_facet Andreev, P.D.
author_sort Andreev, P.D.
title Geometric Constructions in the Class of Busemann Nonpositively Curved Spaces
title_short Geometric Constructions in the Class of Busemann Nonpositively Curved Spaces
title_full Geometric Constructions in the Class of Busemann Nonpositively Curved Spaces
title_fullStr Geometric Constructions in the Class of Busemann Nonpositively Curved Spaces
title_full_unstemmed Geometric Constructions in the Class of Busemann Nonpositively Curved Spaces
title_sort geometric constructions in the class of busemann nonpositively curved spaces
publisher Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate 2009
url http://dspace.nbuv.gov.ua/handle/123456789/106531
citation_txt Geometric Constructions in the Class of Busemann Nonpositively Curved Spaces / P.D. Andreev // Журнал математической физики, анализа, геометрии. — 2009. — Т. 5, № 1. — С. 25-37. — Бібліогр.: 13 назв. — англ.
series Журнал математической физики, анализа, геометрии
work_keys_str_mv AT andreevpd geometricconstructionsintheclassofbusemannnonpositivelycurvedspaces
first_indexed 2025-07-07T18:36:36Z
last_indexed 2025-07-07T18:36:36Z
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fulltext Journal of Mathemati al Physi s, Analysis, Geometry2009, vol. 5, No. 1, pp. 25�37Geometri Constru tions in the Class of BusemannNonpositively Curved Spa esP.D. AndreevLomonosov Pomor State University4 Lomonosov Ave., Arkhangelsk, 163002, RussiaE-mail:pdandreev�mail.ruRe eived May 18, 2007The pro edures of gluing and the Hausdor� limit in the lass of metri spa es nonpositively urved in the sense of Busemann are studied in thepaper. Conditions under whi h the resulting spa es belong to the same lassare found.Key words: Busemann nonpositive urvature, gluing, Hausdor� limit.Mathemati s Subje t Classi� ation 2000: 53C23.1. Introdu tionWe study some onstru tions in the lass of geodesi metri spa es with Buse-mann nonpositive urvature. Here we use not a lassi al de�nition of nonpositively urved spa es ([2℄), but the one introdu ed by B. Bowdit h in [1℄ where the lassof spa es onsidered is alled Busemann. This allows to in lude the onsiderationof all CAT (0)-spa es, i.e., the omplete simply onne ted spa es of nonpositive urvature in the sense of A.D. Alexandrov and all stri tly onvex normed spa es.The main question of the paper is: what are additional onditions for gluing andlimiting operations in the lass of Busemann spa es to keep the resulting spa e inthe same lass?Similar problems for Alexandrov spa es were studied su� iently deeply in[3, 4℄, et . Some operations in the lass of Busemann spa es were studied in [5℄and [6℄. The ondition of nonpositivity of urvature in the sense of Busemannis weaker than in the sense of Alexandrov. By this reason many results that aretrue to Alexandrov spa es do not have dire t generalization for Busemann spa es.When applying Alexandrov spa es theory one should set additional requirementsin a number of situations. P.D. Andreev, 2009 P.D. AndreevThe paper is organized as follows. In Se tion 2 we re all some ne essary de�-nitions and fa ts from Busemann spa es theory. In Se tion 3 we prove the gluingtheorem whi h generalizes Reshetnyak's gluing theorem known for Alexandrovspa es ( f. [3, Th. 9.1.21℄). When we speak about Busemann spa es, the gluingtheorem has the following formulation.Theorem 3.1. Let (X1; d1), (X2; d2) and (X3; d3) be three Busemann spa esrepresented as unions of losed onvex subsets Xi := Ai [ Bi. Let g1 : B2 ! A3,g2 : B3 ! A1 and g3 : B1 ! A2 be three isometries su h that g2Æg1Æg3 = Id jA1\B1 .Then the spa e X obtained as a fa torspa e X := (X1[X2[X3)=fg1; g2; g3g withthe metri d that oin ides with di in ea h Xi, is a Busemann spa e.In Se tion 4 we study the Hausdor� limits of Busemann spa es. The lass ofall Busemann spa es is not losed under Hausdor� limit: the sequen e of stri tly onvex normed spa es an onverge to the normed spa e with nonstri tly onvexnorm. B. Kleiner introdu ed the notion of often onvex spa e in [6℄. The lass ofoften onvex spa es is losed under limits and ontains a sub lass of Busemannspa es. We study the Hausdor� limits of Busemann spa es under additional requi-rement of unimodular onvexity. The main result of the se tion is the followingtheorem.Theorem 4.3. Let the omplete metri spa e (X; o; dX ) with basepoint o bea Hausdor� limit of unimodularly onvex sequen e (Xn; on; dn) of pointed Buse-mann spa es. Then X is also a Busemann spa e and its onvexity modulus Æx(�; r)for all x 2 X is bounded from below by the ommon low boundary of onvexitymodules of spa es Xn. 2. PreliminariesThe general theory of spa es with intrinsi metri an be found in [3, 4℄ and[7℄. Here we re all some basi fa ts related to Busemann nonpositively urvedspa es.De�nition 2.1. Let (X; d) be a geodesi spa e. We use the notation jxyj forthe distan e d(x; y) between its points. A segment onne ting the points x; y 2 Xis denoted [xy℄. We say that X is a Busemann nonpositively urved spa e (shortlyBusemann spa e) if its metri is onvex: if : [a; b℄ ! X and d : [a0; b0℄! X area�ne parameterizations of two segments, then the fun tion D : [a; b℄�[a0; b0℄! R+D(s; t) = j (s)d(t)jis onvex. Equivalently, the spa e X is Busemann nonpositively urved if for anythree points x; y; z 2 X, for the arbitrary midpoint m between x and y and for26 Journal of Mathemati al Physi s, Analysis, Geometry, 2009, vol. 5, No. 1 Geometri Constru tions in the Class of Busemann Nonpositively Curved Spa esarbitrary midpoint n between x and z the inequalityjmnj � 12 jyzj (2.1)holds.The following properties of the onsidered spa es are simple orollaries fromDef. 2.1. Ea h Busemann nonpositively urved spa e X is ontra tible, any itstwo points are onne ted by the unique segment.The lass of Busemann nonpositively urved spa es ontains all CAT (0)-spa es and stri tly onvex Minkowski spa es.The fa t that some non-Minkowskian Finsler manifolds have nonpositive ur-vature in sense of Busemann is less trivial. Finsler metri s having nonpositive urvature in the sense of Busemann were studied in [8℄. It is shown that everyFinsler manifold with Berwald metri and nonpositive �ag urvatures is a genera-lized Busemann spa e (geodesi spa e with the Busemann property of urvaturenonpositivity but without symmetry ondition on the metri ). The Finsler metri F (x; dx) on the manifold Mn is a Berwald metri if there is a spe ial oordinatesystem, where its geodesi s satisfy the system of di�erential equations��i + 2Gi(�; _�) = 0:Here Gi := Gi(x; y) are positive fun tions homogeneous of the se ond degree iny. If the metri F is Riemannian, then Gi = 12�ijk(x)yjyk, where �ijk are Levi�Chivita onne tion oe� ients.The Berwald ondition is essential here. By Kelly�Straus theorem ( f. [9℄),if the Finsler spa e with Hilbert metri (of onstant negative �ag urvature) isa Busemann spa e, then it is a Loba hevsky spa e.In onne tion with onvexity, the spa es with nonpositive urvature are some-times alled onvex spa es ( f. [10℄).De�nition 2.2. The metri spa e X is alled lo ally onvex if every its pointhas a neighborhood that is the Busemann nonpositively urved spa e in the metri of X.Several strengthenings of the onvexity property were introdu ed in [11℄.De�nition 2.3. The Busemann nonpositively urved spa e X is alled stri tly onvex if there is the strong inequalityjx0mj < maxfjx0yj; jx0zjg Journal of Mathemati al Physi s, Analysis, Geometry, 2009, vol. 5, No. 1 27 P.D. Andreevfor every triple of points x0; y; z 2 X, where m is a midpoint between y and z.The stri tly onvex spa e X is alled weakly uniformly onvex if for any pointx0 2 X the modulus of onvexity fun tionÆx0(�; r) := inffr � jx0mj j y; z 2 X;jx0yj � r; jx0zj � r; jyzj � �r; jymj = jmzj = 12 jyzjgis positive for any �; r > 0. Finally, the weakly uniformly onvex spa e X is alleduniformly onvex if limr!+1 Æx0(�; r) = +1for any �xed � > 0.For example, every stri tly onvex Minkowski spa e is uniformly onvex,be ause its modulus of onvexity fun tion is homogeneous by r:Æo(�; �r) = �Æo(�; r)for all �; r; � > 0. 3. GluingThe gluing theorem known for the Alexandrov spa es in Reshetnyak's for-mulation ( f. [12℄) is not true for Busemann spa es with nonpositive urvature.For example, the result of gluing of two normed half-planes with di�erent normsis a plane whose metri fails to be a Busemann nonpositive urvature. We willprove the following version of the gluing theorem.Theorem 3.1. Let (X1; d1), (X2; d2) and (X3; d3) be three Busemann spa esrepresented as unions of losed onvex subsets Xi := Ai [ Bi. Let g1 : B2 ! A3,g2 : B3 ! A1 and g3 : B1 ! A2 be three isometries su h that g2Æg1Æg3 = Id jA1\B1 .Then the spa e X obtained as a fa torspa e X := (X1[X2[X3)=fg1; g2; g3g withthe metri d that oin ides with di in ea h Xi, is a Busemann spa e.P r o f. Identifying ea h spa e Xi with the orresponding subset in X,we noti e that A1 \B1 = B3 \A2 � X1 \X2 \X3:As a orollary, X1 \X2 \X3 = A1 \B1 = A2 \B2 = A3 \B3: 28 Journal of Mathemati al Physi s, Analysis, Geometry, 2009, vol. 5, No. 1 Geometri Constru tions in the Class of Busemann Nonpositively Curved Spa es x y z B =A2 3 B =A3 1 B =A1 2 Fig. 1: The spa e X is a result of the gluing of spa es X1;X2 and X3.Any two points x; y 2 X are ontained in one of Xi and onne ted in thisXi by a segment [xy℄ with natural parametrization : [�; �℄ ! Xi. Assumefor de�niteness that x; y 2 X1. Sin e the distan e in X between the pointsthat belong to X1 oin ides with the distan e d1, then the parametrization isa natural parametrization of the path in the spa e X as well. Consequently,the map represents a segment onne ting x and y in X. It follows that X isa geodesi spa e.Let k =2 X1, that is k 2 B2 n A2 = A3 nB3 = (X2 \X3) nX1, be an arbitrarypoint. Consider the segments [xk℄ and [ky℄ with natural parameterizations p :[�; ℄ ! X and q : [Æ; �℄ ! X. Denote s 2 [�; ℄ the in�mum of parameters� for whi h p(�) =2 X1, and t 2 [Æ; �℄ the supremum of parameters � for whi hq(�) =2 X1. Sin e the sets Ai and Bi are losed, then p(s); q(t) 2 (B2 = A3)\X1.It followsjxyjX = d1(x; y) � d1(x; p(s)) + d1(p(s); q(t)) + d1(q(t); y) < jxkjX + jkyjX :Consequently, every segment onne ting x and y passes in X1, and the pointsx and y are onne ted by the unique segment in X. If x; y 2 Xi, then there isa unique midpoint between x and y and it belongs to the same Xi.Let three points x; y; z 2 X and the midpoints m;n of segments [xy℄ and[xz℄, respe tively, be given. If x; y; z 2 Xi for some i, then also m;n 2 Xi, andthe inequality (2.1) is ful�lled automati ally. Assume that x =2 X1, y =2 X2 andz =2 X3 (as in Fig. 1). Denote p an arbitrary point of the segment [yz℄ in theinterse tion A1 \ B1 = A1 \ A2 \ A3, and q the midpoint of the segment [xp℄(Fig. 2).Journal of Mathemati al Physi s, Analysis, Geometry, 2009, vol. 5, No. 1 29 P.D. Andreev x y z m n p q Fig. 2:Then from x; y; p 2 X3 there follows the inequalityjmqjX � 12 jypjX ; (3.1)and from x; p; z 2 X2 the inequalityjqnjX � 12 jpzjX : (3.2)Combining (3.1) and (3.2), we getjmnjX � jmqjX + jqnjX � 12(jypjX + jpzjX) = 12 jyzjX :4. Convergen e in the Class of Busemann NonpositivelyCurved Spa esDe�nition 4.1. The distortion of the map f : X ! Y of the metri spa e(X; dX ) to the metri spa e (Y; dY ) is de�ned bydis(f) := supx;y2X jdY (f(x); f(y))� dX(x; y)j:The uniform distan e jXY ju between metri spa es (X; dX ) and (Y; dY ) isde�ned by jXY ju := inf dis(f);30 Journal of Mathemati al Physi s, Analysis, Geometry, 2009, vol. 5, No. 1 Geometri Constru tions in the Class of Busemann Nonpositively Curved Spa eswhere the in�mum is taken over all bije tions f : X ! Y . A sequen e (Xn; dn)of metri spa es onverges uniformly to the metri spa e (X; dX ) if jXnXju ! 0.For � > 0; �-net in the metri spa e X is a subset N � X su h that for anyx 2 X there exists a 2 N with jxaj < �:De�nition 4.2 [13, Part I, p. 7℄. Let (X; d) be a bounded metri spa e and(Xi; di) be a sequen e of bounded metri spa es with distan es di. The sequen eXi onverges in the sense of Hausfor� to the spa e X if for any � > 0 there exists�-net N� � X that is a uniform limit of �-nets Ni� in Xi.The de�nition of the Hausdor� onvergen e in the ase of nonbounded spa es isvalid in the ategory of pointed spa es. Let (X; o; d) be a pointed metri spa e withthe marked point o and the metri d, and (Xi; oi; di) be a sequen e of pointed metri spa es with the marked points oi and the metri s di, respe tively. The sequen e Xi onverges in the sense of Hausdor� to the spa e X if for any r > 0 the sequen eof balls BXi(oi; r) onverges in the sense of Hausdor� to the ball BX(o; r).We say that the family of geodesi spa es f(X�; d�)g with the metri s d�is unimodularly onvex if ea h of spa es (X�; d�) is weakly uniformly onvexand there exists the positive fun tion m(�; r) de�ned for �; r > 0 that bounds onvexity modules of all spa es X� from below uniformlyÆx(�; r) � m(�; r) (4.1)for any x 2 X� and for all �.Theorem 4.3. Let the omplete metri spa e (X; o; dX ) with basepoint o isa Hausdor� limit of unimodularly onvex sequen e (Xn; on; dn) of pointed Buse-mann spa es. Then X is also a Busemann spa e and its onvexity modulus Æx(�; r)for all x 2 X is bounded from below by the ommon low boundary of onvexitymodules of spa es Xn.R e m a r k. The unimodular onvexity ondition is essential here.For example, Minkowski planes with the normsk(x; y)kn := npjxjn + jyjnthat are stri tly onvex when n > 1 onverges in the sense of Hausdor� to thenon-stri tly onvex Minkowski plane with maximum normk(x; y)k1 := maxfjxj; jyjg:First, we need the following lemma. Journal of Mathemati al Physi s, Analysis, Geometry, 2009, vol. 5, No. 1 31 P.D. AndreevLemma 4.4. Let the sequen e (Xn; on; dn) and the spa e (X; o; dX ) satisfythe onditions of Th. 4.3. Then:1. X is a geodesi spa e;2. for any two points x; y 2 X the midpoint m between them is unique.P r o f. By Claim 6.1 in [13, Part I℄ the metri of the spa e X is interior.Consequently, X is geodesi as a omplete spa e with interior metri .Now we prove the se ond statement. Assume for the ontrarythat for points x, y 2 X there exists two di�erent midpoints m1, m2 2 X.Put R := 2maxfdX(o; x); dX (o; y)g. From the de�nition of Hausdor� onver-gen e in unbounded spa es, the balls BXn(on; R) onverge to the ball BX(o;R).Let the positive fun tion L(�; r) be de�ned for �; r > 0 by the equalityL(�; r) = inf Æx(�; �);where the in�mum is taken over all x 2 Xn for all natural n. By the inequality(4.1) the in�mum is positive. The fun tion L(�; r) is nonde reasing on � whenr > 0 is �xed. To see this it is su� ient to observe that for all n the fun tionsÆx(�; r) have the mentioned property, where x 2 Xn is arbitrary. Let �2 > �1 > 0.If d�(x; y) � r, d�(x; z) � r and d�(y; z) � �2r hold for the points x; y; z 2 Xn,then also d�(y; z) � �1r. Hen e Æx(�1; r) � Æx(�2; r), and Æ�(�1; r) � Æ�(�2; r).Consequently, for all �; r > 0 there exists � > 0 su h that� < 29L��� 9�r ; r� :Take � > 0 to satisfy the onditionsdX(m1;m2)� 3� > �M(�); (4.2)where M(�) = 12dX(x; y) + 3�, and� < 29L�2dX(m1;m2)� 9�dX(x; y) ; 12dX(x; y)� : (4.3)Let X� be an �-net in the ball B(o;R) � X and a uniform limit of �-nets X�;n inballs B(on; R) � Xn. Let the number N 2 N be taken su h that for all n > Nthere exists a bije tion ��;n : X�;n ! X� for whi hdis��;n < �: (4.4) 32 Journal of Mathemati al Physi s, Analysis, Geometry, 2009, vol. 5, No. 1 Geometri Constru tions in the Class of Busemann Nonpositively Curved Spa esChoose the points x�; y�;m1;�;m2;� 2 X� with the onditionsdX(x; x�) < �;dX(y; y�) < �;dX(m1;m1;�) < �;dX(m2;m2;�) < �:For them dX(x�; y�) � dX(x; y) � 2�;jdX(x�;m1;�)� 12dX(x; y)j � 2�;jdX(x�;m2;�)� 12dX(x; y)j � 2�;jdX(y�;m1;�)� 12dX(x; y)j � 2�;jdX(y�;m2;�)� 12dX(x; y)j � 2�and jdX(m1;�;m2;�)� dX(m1;m2)j � 2�:For arbitrary n > N we havedn(��1�;n(x�); ��1�;n(y�)) � dX(x; y)� 3�; (4.5)and also jdn(��1�;n(x�); ��1�;n(m1;�))� 12dX(x; y)j � 3�; (4.6)jdn(��1�;n(x�); ��1�;n(m2;�))� 12dX(x; y)j � 3�; (4.7)jdn(��1�;n(y�); ��1�;n(m1;�))� 12dX(x; y)j � 3�;jdn(��1�;n(y�); ��1�;n(m2;�))� 12dX(x; y)j � 3�and jdn(��1�;n(m1;�); ��1�;n(m2;�))� dX(m1;m2)j � 3�:Consequently, from (4.2)dn(��1�;n(m1;�); ��1�;n(m2;�)) � �M(�): Journal of Mathemati al Physi s, Analysis, Geometry, 2009, vol. 5, No. 1 33 P.D. Andreev r r rrr r r r r rr rr- - x y�y x� m2m2;�m1zn m1;���1�;n(x�) ��1�;n(y�) ��1�;n(m1;�) ��1�;n(m2;�) ��;n ��;n Fig. 3:Let zn 2 Xn be the midpoint between ��1�;n(m1;�) and ��1�;n(m2;�)(Fig. 3). Consider also the following points. The point p1;n 2 Xn in the segment[��1�;n(x�)��1�;n(m1;�)℄, su h thatdn(��1�;n(x�); p1;n) ==� 12dX(x; y); if dn(��1�;n(x�); ��1�;n(m1;�)) � 12dX(x; y)dn(��1�;n(x�); ��1�;n(m1;�)) otherwise:The point p1;n oin ides with the endpoint ��1�;n(m1;�) of the segment ifdn(��1�;n(x�); ��1�;n(m1;�)) � 12dX(x; y);or its distan e from ��1�;n(x�) is 12dX(x; y) ifdn(��1�;n(x�); ��1�;n(m1;�)) � 12dX(x; y): r r rrr ��1�;n(x�) ��1�;n(m1;�) ��1�;n(m2;�)p1;n p2;nqn Fig. 4:34 Journal of Mathemati al Physi s, Analysis, Geometry, 2009, vol. 5, No. 1 Geometri Constru tions in the Class of Busemann Nonpositively Curved Spa esThe point p2;n is de�ned analogously in the segment [��1�;n(x�)��1�;n(m2;�)℄. Finally,the point qn is the midpoint of the segment [p1;np2;n℄ (Fig. 4).Let us estimate the distan e dn(p1;n; p2;n) from below. From inequalities (4.6)and (4.7) the distan es dn(pi; ��1�;n(mi;�)) and i = 1; 2 satisfy the inequalitydn(pi; ��1�;n(mi;�)) � 3�:Hen e dn(p1;n; p2;n) � dn(��1�;n(m1;�); ��1�;n(m2;�))� 6� � dX(m1;m2)� 9�:We have dn(��1�;n(x�); zn) � dn(��1�;n(x�); qn) + dn(qn; zn)� 12dX(x; y)� L�2dX(m1;m2)� 9�dX(x; y) ; 12dX(x; y)�+12 �dn(p1;n; ��1�;n(m1;�)) + dn(p2;n; ��1�;n(m2;�)�� 12dX(x; y)� L�2dX(m1;m2)� 3�dX(x; y) ; 12dX(x; y)�+ 3�< 12dX(x; y)� 32�:Similarly, dn(��1�;n(y�); zn) < 12dX(x; y)� 32�:Finally, dn(��1�;n(x�); ��1�;n(y�)) < dX(x; y)� 3�; ontradi ting to the inequality (4.5).Now we an omplete the proof.P r o o f of Theorem 4.3. Let the points x; y; z 2 X and the midpoints p andq of segments [xy℄ and [xz℄, respe tively, be given. DenoteR := 2maxfdX(o; x); dX (o; y); dX (o; z)g:Fix the de reasing sequen e �i ! 0.For ea h i, hoose �i-net X�i � BX(o;R) whi h is a uniform limit of �i-netsX�i;n � BXn(o;R). Here BX(o;R) and BXn(on; R) are balls in the spa es X andXn, respe tively.Let n(i) be the natural number su h that there exists a bije tion �i : X�i;n(i) !X�i with the distortion dis�i < �i:Journal of Mathemati al Physi s, Analysis, Geometry, 2009, vol. 5, No. 1 35 P.D. AndreevLet the distan es from the points xi; yi; zi 2 X� to x; y; z, respe tively, be notgreater than �i. Denote ~pi 2 Xn(i) the midpoint of the segment [��1i (xi)��1i (yi)℄,�pi 2 X�i;n(i) the point on the distan e not greater than �i from ~pi, and pi = �i(�pi) 2X its image in the bije tion �i. Sin e by the ondition the spa e X is proper andthe sequen e pi is bounded, one an subtra t the onverging subsequen e. We mayassume that the sequen e pi is also onverging. The points pi are (4�i)-midpointsbetween x and y, that is jdX(x; pi)� 12dX(x; y)j � 4�iand jdX(y; pi)� 12dX(x; y)j � 4�i:Sin e �i ! 0, when i ! 1, the limit of the sequen e pi is the midpoint betweenx and y. From the uniqueness of midpoints in X, it follows thatlimi!1 pi = p:Analogously, one an onstru t the sequen e of (4�i)-midpoints qi between x andz onverging to q. We havedX(pi; qi) � dXn(i)(�pi; �qi) + �i � dXn(i)(~pi; ~qi) + 3�i� 12dXn(i)(��1i (yi); ��1i (zi)) + 3�i � 12dX(yi; zi) + 4�i� 12dX(y; z) + 5�i:Hen e dX(p; q) � 12dX(y; z);that is X is Busemann nonpositively urved. The estimation of the onvexitymodulus Æx(�; r) in X an be proven in a similar way.A knowledgement. The author is grateful to the referee for the number ofimportant remarks and orre tions made.Referen es[1℄ B.H. Bowdit h, Minkowskian Subspa es of Nonpositively Curved Metri Spa es. �Bull. London Math. So . 27 (1995), 575�584.[2℄ H. Busemann, Spa es with Nonpositive Curvature. � A ta Math. 80 (1948), 259�310.36 Journal of Mathemati al Physi s, Analysis, Geometry, 2009, vol. 5, No. 1 Geometri Constru tions in the Class of Busemann Nonpositively Curved Spa es[3℄ D. Burago, Yu. Burago, and S. Ivanov, A Course in Metri Geometry. GraduatedStudies in Mathemati s, 33. AMS, Providen e, RI, 2001.[4℄ M. Bridson and A. Hae�iger, Metri Spa es of Nonpositive Curvature. Comprehen-sive Studies in Mathemati s. 319. Springer-Verlag, Berlin, 1999.[5℄ A. Bernig, Th. Foers h, and V. S hroeder, Non Standard Metri Produ ts. � Contr.Algebra and Geom. 44 (2003), No. 2, 499�510.[6℄ B. Kleiner, The Lo al Stru ture of Length Spa es with Curvature Bounded above.� Math. Z. 231 (1999), 409�456.[7℄ A. Papadopoulos, Metri Spa es, Convexity and Nonpositive Curvature. IRMALe t. in Math. and Theor. Phis. 6. EMS, Strasbourg, 2005.[8℄ A. Krist�alu, C. Varga and L. Kozma, The Dispersing of Geodesi s in Berwald Spa esof Nonpositive Flag Curvature. � Houston J. Math. 30 (2004), No. 2, 413�420.[9℄ P.J. Kelly and E.G. Straus, Curvature in Hilbert Geometry. � Pa i� J. Math. 8(1958), 119�126.[10℄ M. Gromov, Hyperboli Manifolds, Groups and A tions. In: Riemann Surfa es andRelated Topi s. Pro . 1978 Stony Brook Conf. (State Univ. New York, Stony Brook,N.Y., 1978). � Ann. Math. Stud. 97 (1981), Prin eton Univ. Press, Prin eton, NJ,183�213.[11℄ T. Gelander, A. Karlsson, and G. Margulis, Superrigidity, Generalized Harmoni Maps and Uniformly Convex Spa es. � GAFA 17 (2007), 1524�1550.[12℄ Yu.G. Reshetnyak, K Teorii Prostranstv Krivizny ne Bol'shei K. (On the Theory ofSpa es with Curvature not Greater than K). � Mat. Sb., N. Ser. 52 (1960), No. 3,789�798.[13℄ S. Buyalo, Le tures on Spa es of Curvature Bounded above. Illinois, Univ. Urbana-Champaign, Spring Sem. 1994�1995 a.y., Parts I�III. Journal of Mathemati al Physi s, Analysis, Geometry, 2009, vol. 5, No. 1 37

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