bibl.tex

\begin{thebibliography}{52}

\bibitem{ultrafilter} Something with  ultrafilter???

\bibitem{abb:obstacle} Alexander, S.; Berg, D.; Bishop, R., \textit{The Riemannian obstacle problem}.
Illinois J. Math. 31 (1987), 167--184.

\bibitem{a-b-b:CBA-m-w-b} Alexander, S.; Berg, D.; Bishop, R., \textit{ Geometric curvature bounds in Riemannian manifolds with
  boundary}.
  Trans. Amer. Math. Soc., 339 (1993), 703--716.

\bibitem{grad-flow-book} Ambrosio, L.; Gigli, N.; Savar\'e, G., \textit{Gradient flows in metric spaces and in the space of probability measures.} Lectures in Mathematics ETH Z\"urich. Birkh???user Verlag, Basel, 2005. viii+333 pp.

\bibitem{akp} Alexander, S.; Kapovitch, V.; Petrunin A., An optimal lower curvature bound for convex hypersurfaces in Riemannian manifolds ???

\bibitem{akp-kirszbraun}
Alexander, S.; Kapovitch, V.; Petrunin, A.
\textit{Alexandrov meets Kirszbraun.} Proceedings of the G\"okova Geometry-Topology Conference 2010, 88--109, Int. Press, Somerville, MA, 2011. 

\bibitem{a-b:h-c}  
Alexander, S.; Bishop, R., \textit{The Hadamard--Cartan theorem in locally convex spaces}. Enseignement Math., 36 (1990), 309--320.

\bibitem
{a-b:cbc} Alexander, S.; Bishop, R., \textit{Comparison theorems for curves of bounded geodesic curvature
  in metric spaces of curvature bounded above}.
Diff. Geom. Appl. 6 (1996), 67--86.

\bibitem{alexander-bishop:fk} Alexander, S.; Bishop, R., \textit{$\mathcal F
K$-convex functions on metric spaces.} Manuscripta Math. 110, 115133 (2003)


\bibitem{alexander-bishop:warps} Alexander, S. B.; Bishop, R. L. \textit{Curvature bounds for warped products of metric spaces.}  Geom. Funct. Anal.  14  (2004),  no. 6, 1143--1181.

\bibitem{a-b:gauss-equation} 
Alexander, S. B.; Bishop, R. L. \textit{Gauss equation and injectivity radii for subspaces in spaces of curvature bounded above.}  Geometriae Dedicata 117 (2006), 65-84.

\bibitem{alexander-bishop:warp1} Alexander, S. B.; Bishop, R. L. \textit{Warped products admitting a curvature bound.}  Advances in Math.  303  (2016),  88-122.

\bibitem{alexandrov:def} Alexandroff, A.
The inner geometry of an arbitrary convex surface.
C. R. (Doklady) Acad. Sci. URSS (N.S.) 32, (1941). 467--470. 

\bibitem{alexandrov:qg} Alexandrov, A. D., \textit{Curves on convex surfaces,} Doklady Acad. Nauk SSSR v.47 (1945), p.319--322.

\bibitem{alexandrov:int-qg} Alexandrov, A. D., \textit{Quasigeodesics}, Doklady Acad. Nauk SSSR v.69 (1949),
p.717-720.

\bibitem{alexandrov:strong-angle} Aleksandrov, A. D.
A theorem on triangles in a metric space and some of its applications. (Russian) Trudy Mat. Inst. Steklov., v 38, pp. 5--23.


\bibitem{alexandrov:devel} Alexandrow, A. D.,  \textit{\"Uber eine Verallgemeinerung der Riemannschen Geometrie,}
Schriftenreihe Inst. Math. 1 (1957), p.33--84.

\bibitem{alexandrov:ruled} Alexandrov, A. D., \textit{Ruled surfaces in metric spaces,}
Vestnik Leningrad Univ. 12  (1957), 5--26.

\bibitem{alexandrov-embedding} Alexandrov, A. D.???

\bibitem{alexandrov-glueing} Alexandrov, A. D???

\bibitem{alexandrov-burago} Alexandrov, A. D.; Burago, Yu. D., \textit{Quasigeodesics}, Trudy Mat. Inst. Steklov.
76 (1965), p.49--63.

\bibitem{ADG}Ancel, F. D.; Davis, M. W.; Guilbault, C. R.
CAT(0) reflection manifolds. Geometric topology, 441--445,
AMS/IP Stud. Adv. Math., 2.1, Amer. Math. Soc., Providence, RI, 1997.

\bibitem{ballmann:cartan-hadamard}
Ballmann, W., \textit{Singular spaces of non-positive curvature}, Sur les Groupes Hyperboliques d'apr\'{e}s Mikhael Gromov (Ghys, E., de la Harpe, P., ed.), Chapter 10, Progr. Math. 83, Birkh\"auser, Boston, 1990. 


\bibitem{ballmann:lectures}
Ballmann, W., \textit{Lectures on Spaces of Nonpositive Curvature}, DMV Seminar 25, Birkh\"auser Verlag, Basel, 1995. 

\bibitem{ballmann:notes}
Ballmann, W., \textit{On the Geometry of Metric Spaces}, http:/???

\bibitem{berestovskii} Berestovskij, V. 
\textit{Borsuk's problem on metrization of a polyhedron,}  
Dokl. Akad. Nauk. SSSR 27 (1983), 56--59.

%\bibitec{berestovskii:quadruples} Berestovskii, V. 
%\textit{Spaces with bounded curvature and distance geometry,}  
%Sib. Math. J. 24 (1986), 8--19;
%translation from Sib. Mat. Zh. 27:1 (1986), 11--25.

\bibitem{besicovitch-moran}
Besicovitch, A. S.; Moran, P. A. P.
\textit{The measure of product and cylinder sets.}
J. London Math. Soc. 20, (1945). 110--120. 

\bibitem{BHV} Billera, Louis J.; Holmes, Susan P.; Vogtmann, Karen Geometry of the space of phylogenetic trees. Adv. in Appl. Math. 27 (2001), no. 4, 733?767.

\bibitem{bishop:jordan}Bishop, Richard L. The intrinsic geometry of a Jordan domain. Int. Electron. J. Geom. 1 (2008), no. 2, 33?39.

\bibitem{bowditch} Bowditch, B. H.,
\textit{Notes on locally CAT(1) spaces} 
in ``Geometric group theory'', (ed. R. Charney, M. Davis, M. Shapiro), de Gruyter (1995) 1--48.

\bibitem{BH}
Bridson, M., Haefliger, A.,
\textit{Metric Spaces of Non-positive Curvature},
Springer-Verlag, Berlin,1999.c
\bibitem{blaschke} Blaschke, W., \textit{Kreis und Kugel} Leipzig: Veit, 1916.

\bibitem{BBI} Burago, D.; Burago, Yu.; Ivanov, S., \textit{A course in metric geometry.} Graduate Studies in Mathematics, 33. American Mathematical Society, Providence, RI, 2001. xiv+415 pp. ISBN: 0-8218-2129-6

\bibitem{BFK}Burago, D., Ferleger, S., Kononenko, A.,
\textit{Uniform estimates on the number of collisions in semi-dispersing
billiards},
Annals of Math.(2) 147 (1998), 695-708.


\bibitem{BGP} Burago, Yu.; Gromov, M.; Perelman, G., \textit{A. D. Aleksandrov spaces
with curvatures bounded below.} (Russian)  Uspekhi Mat. Nauk  47  (1992),  no.
2(284), 3--51, 222;  translation in  Russian Math. Surveys  47  (1992),  no. 2, 1--58

\bibitem{busemann-CBA}Busemann, Herbert 
Spaces with non-positive curvature. Acta Math. 80, (1948). 259--310.


\bibitem{busemann} Busemann, H., \textit{Convex Surfaces}, Tracts in Pure and Appl. Math. 6, Interscience, New York, 1958.

\bibitem{buser-karcher} Buser, P.; Karcher, H.,
Gromov's Almost Flat Manifolds,
Soc. Math. de France, Ast\'erisque Fascicules
81 (1981). 

\bibitem{buyalo:convex-surface} Buyalo, S., Shortest paths on convex
hypersurface of a Riemannian manifold (Russian), Studies in Topology, Zap. Nauchn. Sem.
LOMI 66 (1976) 114--132; translated in J. of Soviet Math. 12 (1979), 73--85.

\bibitem{buyalo:lecture}
S. Buyalo,
{\em Lecture notes on spaces of nonpositive curvature},
www.pdmi.ras.ru/buyalo/add.html.

\bibitem{cartan}Cartan, \'E.
Le\c{c}ons sur la G\'eom\'etrie des Espaces de Riemann. 
Gauthier-Villars, Paris, 1928. vi+273 pp.


\bibitem{cheeger-gromoll-split}
Jeff Cheeger; Detlef Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature, Journal of Differential Geometry 6 (1971/72), 119--128.

\bibitem{cheeger-gromoll-soul}  Cheeger, J.;  Gromoll, D., \textit{On the structure of complete manifolds of non-negative curvature,} Ann. of Math. 96 (1972), 413--443.


\bibitem{cohn-vossen_line} S. Cohn-Vossen: Totalkr\"ummung und geod\"atische Linien auf einfach zusammenh\"angenden, offenen, vollst\"andigen Fl\"achenst\"ucken. Math. Sb.43 (1936), 139--163

\bibitem{daniilidis-ley-sabourau}
Daniilidis, A.; Ley, O.; Sabourau, S.,
\textit{Asymptotic behaviour of self-contracted planar curves and gradient orbits of convex functions.}
J. Math. Pures Appl. (9) 94 (2010), no. 2, 183--199. 

\bibitem{danzer-gruenbaum}
Danzer, L., Gr\"unbaun, B., \textit{???} ``Math. Z.'',
1962, Bd 79, S. 95--99.

\bibitem{davis-noneuclidean}
Davis, Michael W., Groups generated by reflections and aspherical manifolds not covered by Euclidean space.
Ann. of Math. (2) 117 (1983), no. 2, 293--324. 

\bibitem{davis-survey} Davis, Michael W. Exotic aspherical manifolds. Topology of high-dimensional manifolds, No. 1, 2 (Trieste, 2001), 371--404, ICTP Lect. Notes, 9, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2002.

\bibitem{davis-januszkiewicz} Davis, Michael W.; Januszkiewicz, Tadeusz Hyperbolization of polyhedra. J. Differential Geom. 34 (1991), no. 2, 347--388.

\bibitem{davis-januszkiewicz-lafont} 
Davis, M.; 
Januszkiewicz, T.; 
Lafont, J.-F.
4-dimensional locally CAT(0)-manifolds with no Riemannian smoothings. 
Duke Math. J. 161 (2012), no. 1, 1--28. 

\bibitem{edwards} Edwards, Robert D.
The topology of manifolds and cell-like maps. Proceedings of the International Congress of Mathematicians (Helsinki, 1978), pp. 111--127, Acad. Sci. Fennica, Helsinki, 1980.

\bibitem{erdos} Erd\H{o}s, P. \textit{???} Michigan Math. J., 1957, no. 4,
p. 291--300.

\bibitem{eshenburg-split}, J.-H. The splitting theorem for space-times with strong energy condition.

\bibitem{engelking} Engelking, R., \textit{Dimension theory.} Translated from the Polish and revised by the author. North-Holland Mathematical Library, 19. North-Holland Publishing Co., Amsterdam-Oxford-New York; PWN---Polish Scientific Publishers, Warsaw, 1978. x+314 pp.

\bibitem{federer}Federer, H., \textit{Geometric measure theory} ???

\bibitem{schroeder-foetch}
Foertsch, T;  Schroeder, V. \textit{Minkowski versus Euclidean rank for products of metric spaces.}  Adv. Geom.  2  (2002),  no. 2, 123--131.

\bibitem{FY}Fukaya, K.; Yamaguchi, T., \textit{Almost nonpositively curved manifolds.}  J. Differential Geom.  33  (1991),  no. 1, 67--90.

\bibitem{frolik} Frol\'{\i}k, Z., \textit{Concerning topological convergence of sets}, Czechoskovak Math. J. 10 (1960), 168--180.

\bibitem{goffman} Goffman, C.,
\textit{Real Functions}, Rinehart, New York,1953.

\bibitem{greene-wu} Greene, R. E.; Wu, H.-H., \textit{On the subharmonicity and plurisubharmonicity of geodesically convex
functions}.
Indiana Univ. Math. J. 22 (1972/73), 641--653.

\bibitem{grove-petersen:rad-sphere}  Grove, K.; Petersen, P. \textit{A radius sphere theorem,}
Invent. Math. 112 (1993), 577--583.

\bibitem{gromov-polynomial-growth}Gromov, M.???

\bibitem{gomov-precompactness} Gromov, M.???

\bibitem{gromov-MS} Gromov, M., Metric Structures...

\bibitem{gromov-filling}  Gromov, M., \textit{Filling Riemannian manifolds,} J. Diff. Geo. 18 (1983), 1--148.


\bibitem {gromov:hyp-groups}
Gromov. M.,
Hyperbolic groups, 
Essays in Group Theory (S.M. Gersten, ed.),
Math. Sciences Research Institute Publications 8,
Springer-Verlag, Berlin,
1987, 75--264.

\bibitem{gromov:asymt-inv}
Gromov, M., Asymptotic invariants for infinite groups. 
In Niblo and Roller, eds.,
Geometric group theory. London Mathematical Society 1993.

\bibitem{gromov:SaGMC}Gromov, M.(F-IHES)
Sign and geometric meaning of curvature. 
Rend. Sem. Mat. Fis. Milano 61 (1991), 9--123 (1994). 

\bibitem{gromov-CAT} Gromov, M., \textit{CAT($\kappa$)-spaces: Construction and Concentration,} Journal of Mathematical Sciences, Vol. 119, No. 2, (2004) 178--200.



\bibitem{grove-petersen:finiteness} Grove, K.; Petersen, P., V
\textit{Bounding homotopy types by geometry.}
Ann. of Math. (2) 128 (1988), no. 1, 195--206.

\bibitem{grove-wilking} Grove, K.; Wilking, B.
A knot characterization and 1-connected nonnegatively curved 4-manifolds with circle symmetry. 
Geom. Topol. 18 (2014), no. 5, 3091--3110.

\bibitem{hadamard} Hadamard, J.;
Sur la forme des lignes g\'eod\'esiques \'a l'infini et sur les g\'eod\'esiques des surfaces r\'egl\'ees du second ordre.
Bull. Soc. Math. France 26 (1898), 195??216.


\bibitem{halbeisen} Halbeisen, S., \textit{On tangent cones of Alexandrov spaces with curvature
bounded below.}  Manuscripta Math.  103  (2000),  no. 2, 169--182.

\bibitem{hausdorff}  Hausdorff, F. Grundz\"uge der Mengenlehre. von Veit, Leipzig, 1914. viii + 476pp.

\bibitem{heinonen}
Heinonen, J.
Lectures on Analysis on Metric Spaces,
2001


\bibitem{hitzelberger-lytchak}
Hitzelberger, P.; Lytchak, A.,
Spaces with many affine functions. 
Proc. Amer. Math. Soc. 135 (2007), no. 7, 2263--2271.


\bibitem{top-dim} Hurewicz, W.; Wallman, H., \textit{Dimension theory,} 1941;
???  Dimension Theory. Princeton University Press 1969,
       Princeton Mathematical Series
       
\bibitem{hsiang-kleiner}
Hsiang, Wu-Yi; Kleiner, Bruce
On the topology of positively curved 4-manifolds with symmetry.
J. Differential Geom. 29 (1989), no. 3, 615--621. 

\bibitem{ivanov:local-global-convexity}
Nathan Reading (http://mathoverflow.net/users/5519/nathan-reading), To what extent is convexity a local property?, URL (version: 2010-04-21): http://mathoverflow.net/q/22062

\bibitem{isbell} Isbell, J. \textit{Six theorems about metric spaces}. Comment. Math. Helv. 39: 65--74.(1964).



\bibitem{jost-ball} Jost, J.,
Eine geometrische Bemerkung zu S\"atzen \"uber
harmonische Abbildungen, die ein Dirichletproblem l\"osen,
Manus. Math. 32  (1980),  51--57.

\bibitem{jost} Jost, J., Nonlinear Dirichlet forms??? 1998

\bibitem{jun:thesis} Jun, C. Pursuit-evasion and time-dependent gradient flow in singular spaces,
thesis at University of Illinois at Urbana-Champaign, 2012.

\bibitem{jun:grad} Jun, C., Time-dependent gradient curves on CAT(0) spaces, \texttt{arXiv:1801.04583 [math.MG]}

\bibitem{kapovitch:regularity}Kapovitch, V.???

\bibitem{kapovitch:collapsing}Kapovitch, V.???

\bibitem{kapovitch:stability} Kapovitch, V.,  Perelman's stability theorem. ???

\bibitem{kervaire} Kervaire, M. , "Smooth homology spheres and their fundamental groups", Transactions of the American Mathematical Society ?. 144 (1969): 67--72

\bibitem{kirchheim} Kirchheim, B.,  \textit{Rectifiable metric spaces: local structure and regularity of the Hausdorff measure.} Proc. Amer. Math. Soc. 121 (1994), 113--123.

\bibitem{kirszbraun} Kirszbraun, M. D., \textit{\"Uber die zusammenziehende und Lipschitzsche Transformationen,} Fundamenta Math. 22 (1934), 77--108.

\bibitem{kleiner-leeb}  Kleiner, B.; Leeb, B., \textit{Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings.} Inst. Hautes \'Etudes Sci. Publ. Math. No. 86 (1997), 115--197 (1998).

\bibitem{kleiner} Kleiner, B.,  \textit{The local structure of length spaces
with curvature bounded above,} Math. Z. 231, 409--456 (1999)

\bibitem{kopecka-reich}Kopeck\'a, E.; Reich, S.,
Nonexpansive retracts in Banach spaces. (English summary) Fixed point theory and its applications, 161--174, 
Banach Center Publ., 77, Polish Acad. Sci. Inst. Math., Warsaw, 2007. 

\bibitem{kosovskii-cbb}
Kosovski, N.~N. 
\textit{Gluing of Riemannian manifolds of curvature $\geq \kappa$}, Algebra i Analiz, 14(3) (2002), 140--157.

\bibitem{kosovskii-cba1}
Kosovski, N.~N. 
\textit{Gluing of Riemannian manifolds of curvature $\leq \kappa$}, Algebra i Analiz, 14(3) (2002), 140--157.

\bibitem{kosovskii-cba2}
\textit{Gluing with branching of {R}iemannian manifolds of curvature $\leq \kappa$}

\bibitem{KPT} Kapovitch, V.; Petrunin, A.; Tuschmann, W.,  \textit{Nilpotency, Almost Nonnegative Curvature and the Gradient Push,} to appear in Annals of Mathematics.

\bibitem{kuczumow--stachura} Kuczumow, T.; Stachura, A., Extensions of nonexpansive mappings in
the Hilbert ball with the hyperbolic metric I, II, Comment. Math. Univ.
Carolin. 29 (1988), 399--410.

\bibitem{lang-schroeder}  Lang, U.; Schroeder, V., \textit{Kirszbraun's theorem and metric spaces of bounded curvature,} Geom. funct. anal.
Vol. 7 (1997) 535--560

\bibitem{lebedeva} Lebedeva, Nina ???Erd\H{o}s' problem for Alexandrov's spaces???

\bibitem{lebedeva-curv} Lebedeva, Nina, Curvature of smoothable Alexandrov spaces ??? 

\bibitem{lebedeva-petrunin} Lebedeva, Nina; Petrunin, Anton 
Curvature bounded below: a definition a la Berg--Nikolaev. 
Electron. Res. Announc. Math. Sci. 17 (2010), 122--124.

\bibitem{lebedeva-petrunin-poly} Lebedeva, Nina; Petrunin, Anton, 
Local characterization of polyhedral spaces,
{\tt arXiv:1402.6670}

\bibitem{LPZ} Lebedeva, Nina; Petrunin, Anton and Zolotov Vladimir
Tree comparison (in preparation)

\bibitem{li}Li, N.,
\textit{Volume and gluing rigidity in Alexandrov geometry}
arXiv:1110.5498

\bibitem{liberman}Liberman, J. \textit{Geodesic lines on convex surfaces.}  C. R.
(Doklady) Acad. Sci. USSR (N.S.)  32,  (1941). 310--313.

\bibitem{lytchak:diff}Lytchak, A. \textit{Differentiation in metric spaces.}  Algebra i Analiz, Vol.16 (2004), 128-161 

\bibitem{lytchak:open-map}Lytchak, A. \textit{Open map theorem for metric spaces.}  Algebra i Analiz  17  (2005),  no. 3, 139--159;  translation in  St. Petersburg Math. J.  17  (2006),  no. 3, 477--491 

\bibitem{lytchak-affine} Lytchak, A. \textit{Affine maps???}

\bibitem{mangoldt} Hans von Mangoldt 
\textit{Ueber diejenigen Punkte auf positiv gekr\"ummten Fl\"achen, welche die Eigenschaft haben, dass die von ihnen ausgehenden geod\"atischen Linien nie aufh\"oren, k\"urzeste Linien zu sein.} // J. Reine Angew. Math. --- 1881. --- Bd. 91. --- S. 23--53.

\bibitem{mayer} Mayer, U. F. Gradient flows on nonpositively curved metric spaces and harmonic maps. Comm. Anal. Geom. 6 (1998), no. 2, 199--253.

\bibitem{mazur}Mazur, Barry A note on some contractible 4-manifolds. Ann. of Math. (2) 73 1961 221--228. 

\bibitem{mendonca:shioya}
Mendon\c{c}a, S., \textit{The asymptotic behavior of the set of rays,} Comment. Math. Helv. 72 (1997) 331--348.


\bibitem{menger:wald} Menger, K., \textit{The formative years of Abraham Wald and his work in geometry}, Ann. Math. Statistics 23 (1952), 14--20.


\bibitem{milka-line}Milka, A. D., \textit{Metric structure of a certain class of spaces that contain straight lines.} (Russian) Ukrain. Geometr. Sb. Vyp.  4  1967 43--48.

\bibitem{milka:poly1}Milka, A. D., \textit{Multidimensional spaces with polyhedral metric of nonnegative curvature. I.}  (Russian)  Ukrain. Geometr. Sb. Vyp.  5--6 
1968 103--114. 

\bibitem{milka:poly2}Milka, A. D., \textit{Multidimensional spaces with polyhedral metric of nonnegative curvature. II.}  (Russian)  Ukrain. Geometr. Sb. No. 7
(1969), 68--77, 185 (1970).

\bibitem{milka:qg}Milka, A. D. \textit{Certain properties of quasigeodesics.} 
Ukrain. Geometr. Sb. No. 11 (1971), 73--77.

\bibitem{milka-conv} {Milka, A. D.} {Shortest arcs on convex surfaces} (Russian),  {Dokl. Akad.
Nauk SSSR}  {248}  {1979}, {no. 1}, {34--36};  translated in  Soviet Math. Dokl. 20 (1979), 949--952.

\bibitem{milka-parallel-conv} Milka, A. D.  \textit{Geodesics and shortest lines on convex hypersurfaces.  I.}  (Russian)   Ukrain. Geometr. Sb. No. 25
(1982), 95 - 110, 143 (1982).

\bibitem{monod} Monod, N. \textit{Superrigidity for irreducible lattices and geometric splitting,} J. of the AMS,
Vol. 19, Num. 4, October 2006, 781--814

\bibitem{nagata} Nagata, Jun-iti \textit{Modern dimension theory.} Revised edition. Sigma Series in Pure Mathematics, 2. Heldermann Verlag, Berlin, 1983. ix+284 p.

\bibitem{nikolaev} Nikolaev, I. G. \textit{On the parallel displacement of vectors in spaces with
bilaterally bounded curvature in the sense of A.D. Aleksandrov}, Sib. Math.
J. 24 (1983), 106--119; translation from Sib. Mat. Zh. 24:1(137) (1983), 130--145.


\bibitem{ohta} Ohta, S., \textit{Gradient flows on Wasserstein spaces
over compact Alexandrov spaces} ???

\bibitem{otsu:second-der} Otsu, Yukio Differential geometric aspects of Alexandrov spaces.  Comparison
geometry (Berkeley, CA, 1993--94),  135--148, Math. Sci. Res. Inst. Publ., 30,
Cambridge Univ. Press, Cambridge, 1997.

\bibitem{otsu-shioya} Otsu, Y.; Shioya, T., \textit{The Riemannian structure of Alexandrov spaces.}  J. Differential Geom.  39  (1994),  no. 3, 629--658.

\bibitem{panov-petrunin:sweeping}
Panov, Dmitri; Petrunin, Anton
Sweeping out sectional curvature. 
Geom. Topol. 18 (2014), no. 2, 617--631. 

\bibitem{panov-petrunin:ramification}
Panov, Dmitri; Petrunin, Anton
Ramification conjecture and Hirzebruch's property of line arrangements

\bibitem{papadopoulos}  Papadopoulos, A.
\textit{Metric Spaces, Convexity and Nonpositive Curvature},
IRMA Lectures in Mathematics and Theoretical Physics 6, European Mathematical Society, 2005. 


\bibitem{perelman:spaces2} Perelman, G. \textit{Alexandrov paces with curvature
bounded from below II.} Preprint LOMI, 1991. 35pp.

\bibitem{perelman:morse}Perelman, G. Ya., \textit{Elements of Morse theory on
Aleksandrov spaces.} (Russian)  Algebra i Analiz  5  (1993),  no. 1, 232--241; 
translation in  St. Petersburg Math. J.  5  (1994),  no. 1, 205--213

\bibitem{perelman-soul}Perelman, G???

\bibitem{perelman:collapsing} Perelman, G. \textit{Collapsing with no proper extremal subsets.}  Comparison geometry (Berkeley, CA, 1993--94),  149--155,
Math. Sci. Res. Inst. Publ., 30, Cambridge Univ. Press, Cambridge, 1997.


\bibitem{perelman:DC} Perelman, G., \textit{DC Structure on Alexandrov Space,}\\ \href{http://www.math.psu.edu/petrunin/papers/papers.html}{http://www.math.psu.edu/petrunin/}

\bibitem{perelman-petrunin:extremal} Perelman, G. Ya.; Petrunin, A. M.
\textit{Extremal subsets in Aleksandrov spaces and the generalized Liberman theorem.}
(Russian)  Algebra i Analiz  5  (1993),  no. 1, 242--256;  translation in  St.
Petersburg Math. J.  5  (1994),  no. 1, 215--227

\bibitem{perelman-petrunin:qg} Perelman, G.; Petrunin,  A., \textit{Quasigeodesics and Gradient curves in Alexandrov spaces.}
 \href{http://www.math.psu.edu/petrunin/papers/papers.html}{http://www.math.psu.edu/petrunin/}

\bibitem{petersen:list} Petersen, P. \textit{Comparison geometry problem list.} Riemannian geometry (Waterloo, ON, 1993), 87--115,
Fields Inst. Monogr., 4, Amer. Math. Soc., Providence, RI, 1996. 

\bibitem{petersen:RiemGeom} Petersen, P., Riemannian Geometry,
Springer-Verlag, New York Berlin Heidelberg, 1998.

\bibitem{petrunin:master}Petrunin, A.???

\bibitem{petrunin:extremal} Petrunin, A., \textit{Applications of quasigeodesics
and gradient curves.}  Comparison geometry (Berkeley, CA, 1993--94),  203--219,
Math. Sci. Res. Inst. Publ., 30, Cambridge Univ. Press, Cambridge, 1997.

\bibitem{petrunin:parallel} Petrunin, A., \textit{Parallel transportation for Alexandrov space with curvature bounded below.}  GAFA, Vol. 8 (1998) 123--148

\bibitem{petrunin:PL} Petrunin, A. ???

\bibitem{petrunin:survey} Petrunin, A.,  Semiconcave Functions in Alexandrov's Geometry, Surveys in Differential Geometry XI.

\bibitem{petrunin:optimal} Petrunin, A., Alexandrov meets Lott--Villani--Sturm. M\"unster J. Math. 4 (2011), 53--64.

\bibitem{phelps}Phelps, R. R., Convex functions, monotone operators and differentiability, LNM 1364

\bibitem{plaut:dimension} {Plaut, C.},
      \textit{{Spaces of Wald--Berestovskii curvature bounded below}},
    {J. Geom. Anal.},
     {6},
     {1996},
     {1},
      {113--134},

\bibitem{plaut:survey}Plaut, C., \textit{Metric spaces of curvature $\ge k$.} 
Handbook of geometric topology,  819--898, North-Holland, Amsterdam, 2002.

\bibitem{pogorelov:qg} Pogorelov, A. V., \textit{Qusigeodesic lines on a convex surface,} Mat. Sb. v. 25/2
(1949), p.275-306.

\bibitem{pogorelov:book}  Pogorelov, A. V., 
\textit{Extrinsic geometry of convex surfaces}, 
Transl. Math. Monographs 35, 
American Mathematical Society, Providence, RI, 1973.


\bibitem{pontyagin-surface} L. Pontrjagin \textit{Sur une hypothese fondamentale de la theorie de la dimension} Gomptes Rendus.  1930.  190. p. 1105--1107.

\bibitem{reshetnyak:glue}
Reshetnyak, Yu. G.,
\textit{On the theory of spaces of curvature not greater than $K$},
 Mat. Sb. 52 (1960), 789 -- 798 (Russian).
English translation: \textit{Inextensible mappings in a space of curvature
no greater than $K$}, Siberian Math. Jour. 9 (1968),683-689.

\bibitem{reshetnyak:major}
Reshetnyak, Yu. G.,
\textit{Nonexpanding maps in a space of curvature no greater than $K$},
Sibirskii Mat. Zh. 9 (1968), 918-928 (Russian).
English translation: \textit{Inextensible mappings in a space of curvature
no greater than $K$}, Siberian Math. Jour. 9 (1968),683-689.


\bibitem{riemann} Riemann, B., \textit{\"Uber die Hypothesen welche der Geometrie zu Grunde liegen,}
Abhandlungen der K\"oniglichen Gesellschaft der Wissenschaften zu G\"ottingen, vol. 13, 1867

\bibitem{rinow}Rinow, W. Die innere Geometrie der metrischen Raume. Springer, Berlin, Geidelberg, New York, 1961.

\bibitem{rudin}
Rudin, W.,
\textit{Homogeneity problems in the theory of \v{C}ech compactifications.}
Duke Math. J. 23 (1956), 409?419. 

\bibitem{sevar'e}Savar\'e, G, \textit{Gradient flows and diffusion semigroups in metric spaces under lower curvature bounds.}  C. R. Math. Acad. Sci. Paris  345  (2007),  no. 3, 151--154. 

\bibitem{SW} E. Sawyer and R. L. Wheeden, Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces, Amer. J. Math. 114 (1992), no. 4, 813?874.

\bibitem{shankar-sormani}Shankar, K.; Sormani, C., \textit{Conjugate points in length spaces,} 
Adv. Math. 220 (2009), 791--830.

\bibitem{shefel-2D} S. Z. \v{S}efel?, On saddle surfaces bounded by a rectifiable curve, Dokl. Akad. Nauk SSSR 162 (1965), 294--296.

\bibitem{shefel-3D} S. Z. \v{S}efel?, On the intrinsic geometry of saddle surfaces, Sibirsk. Mat. ?. 5 (1964), 1382--1396

\bibitem{shiohama} Shiohama, K., \textit{An introduction to the geometry of
Alexandrov spaces.} Lecture Notes Series, 8. Seoul National University, Research
Institute of Mathematics, Global Analysis Research Center, Seoul, 1993. ii+78
pp.

\bibitem{shioya}  Shioya, T., \textit{Splitting theorems for nonnegatively curved open manifolds with large ideal boundary,} Math. Zeit. 212 (1993), 223--238.

\bibitem{sharafutdinov}Sharafutdinov, V. A., 
\textit{The Pogorelov--Klingenberg theorem for manifolds homeomorphic to $\RR^n$,} 
Sib. Math. J. v.18/4 (1977),  915--925.

\bibitem{stone}Stone, David A.
Geodesics in piecewise linear manifolds.
Trans. Amer. Math. Soc. 215 (1976), 1--44. 

\bibitem{sturm}Sturm, K. T. \textit{Metric spaces of lower bounded curvature.}  
Exposition. Math.  17  (1999),  no. 1, 35--47.

\bibitem{toponogov-globalization+splitting}  Toponogov, V. A. \textit{Riemannian spaces of curvature bounded below,} Usp. Mat. Nauk 14 (1959), 87--130.

\bibitem{valentine-sphere} Valentine F.A., 
\textit{Contractions in non-Euclidean spaces}, Bull. Amer. Math. Soc. 50 (1944), 710--713.

\bibitem{valentine-kirszbraun} Valentine, F. A., 
\textit{A Lipschitz condition preserving extension for a vector
function}, Amer. J. Math. 67 (1945), 83--93.

\bibitem{villani} Villani, C., \textit{Optimal transport, old and new,} Grundlehren der mathematischen Wissenschaften, Vol. 338, Springer, 2008.

\bibitem{vilms} Vilms, J., \textit{Totally geodesic maps}.
J. Differential Geometry 4 1970 73--79. 

\bibitem{wald} Wald, A., \textit{Begr\"undung eiiner Koordinatenlosen Differentialgeometrie der Fl\"achen}, Ergebnisse eines mathematischen Kolloquium
6  (1935), 24--46.



\bibitem{wijsman} Wijsman, R. A. Convergence of sequences of convex sets, cones and functions. II. Trans. Amer. Math. Soc. 123 1966 32--45.

\bibitem{yamaguchi:bundle} Yamaguchi, T., \textit{Collapsing and pinching under a lower curvature
bound,} Ann. of Math. (2) 113 (1991), 317--357.

\bibitem{zajicek} Zaj\'{\i}\v{c}ek, L. \textit{On the differentiation of convex functions in finite and infinite dimensional spaces.} Czechoslovak Mathematical Journal, vol. 29 (1979), issue 3, pp. 340--348
\begin{thebibliography}{52}