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19.12.2012. Topics in the quasihyperbolic geometry in Banach spaces, 18 pages. pdf

Abstract: Still changing collection of miscellaneous unpublished results and background material.

17.12.2012. Tangential properties of quasihyperbolic geodesics in Banach spaces, 14 pages. pdf

Abstract: In a large class of Banach spaces, the quasihyperbolic geodesics are smooth.

24.3.2012. Regular simplices in three-dimensional normed spaces, 2 pages, Beitr. Alg. Geom. 52, 2012, 569-570. pdf

Abstract: A new proof for the theorem of Petty: Every 3-dimensional normed space contains a regular 3-simplex.

15.7.2011. Slopes of bisectors in normed planes, 9 pages, to appear in Beitr. Alg. Geom. pdf

Abstract: Study of the metric properties of the bisector of a pair of points in a normed plane.

5.10.2008. Quasihyperbolic geometry of planar domains, 30 pages, Ann. Acad. Sci. Fenn. Math. 34, 2009, 447-473. pdf

Abstract: QH disks of radius < 1 are strictly convex and other basic results in planar QH geometry.

6.11.2007. Holes of maps of euclidean domains, 10 pages, Conform. Geom. Dyn. 12, 2008, 58-66. pdf

Abstract: Behavior of the quasiconvexity and bounded turning of holes of domains under quasisymmetric and bilipschitz maps.

23.9.2006 Quasihyperbolic geometry of domains in Hilbert spaces, 24 pages, Ann. Acad. Sci. Fenn. Math. 32, 2007, 559-578. pdf

Abstract: Basic smoothness and bilipschitz properties of geodesics, balls and spheres in the quasihyperbolic metric.

7.6.2006 Quasihyperbolic geodesics in convex domains II (together with O.Martio), 12 pages, Pure Appl. Math. Q. 7, 2011, 379-393. pdf

Abstract: Continuation of the previous part.

3.4.2006 Quasihyperbolic geodesics in convex domains, 12 pages, Result. Math. 48, 2005, 184-195. pdf

Abstract: QH geodesics exist in convex domains in reflexive Banach spaces and they are quasiconvex.

2.2.2004 Broken tubes in Hilbert spaces, 12 pages, Analysis 24, 2004, 227-238. pdf

Abstract: A broken tube is a domain in a Hilbert space. It has several properties that do not occur in finite dimensions.

4.5.2004 Hyperbolic and uniform domains in Banach spaces, 44 pages, Ann. Acad. Sci. Fenn. Math. 30, 2005, 261-302. pdf

Abstract: Generalizing results of Bonk-Heinonen-Koskela I study relations between uniform and Gromov hyperbolic domains in arbitrary Banach spaces.

3.5.2004 Gromov hyperbolic spaces, 42 pages. Expo. Math. 23, 2005, 187-231. pdf

Abstract: A mini monograph on Gromov hyperbolic spaces, which need not be geodesic or proper.

4.11.2003 Linear bilipschitz extension property (together with P. Alestalo and D.A. Trotsenko), Sibirsk. Mat. Zh. 44, 2003, 959-968. pdf

Abstract: We give a sufficient quantitative geometric condition for a subset A of the n-space to have the following property for a given C > 1: There is t > 0 such that for 0 < s < t, each (1+s)-bilipschitz map f of A into the n-space has an extension to a (1+Cs)-bilipschitz map of the whole n-space.

2002 Locally bilipschitz maps of roughly dense sets, Analysis 22, 2002, 437-444. pdf

Abstract: If f is a map of a roughly dense subset of R^n into R^n and if f is L-bilipschitz in sufficiently large balls, then f is L-bilipschitz.

2002 A proof of the Mazur-Ulam theorem, Amer. Math. Monthly 110, 2003, 634-636. pdf

Abstract: In this easy proof no sequence is constructed.

2002 Nonsurjective nearisometries of Banach spaces (together with P. Semrl), J. Funct. Anal. 198, 2003, 268-278. pdf

Abstract: The Hyers-Ulam-Gevirtz theorem holds for nearsurjective nearisometries between Banach spaces.

2001 Wall properties of domains, Ann. Acad. Sci. Fenn. Math. 27, 2002, 437-444. pdf

Abstract: We improve the wall theorem for domains in a euclidean space by replacing the euclidean metric by the inner metric of the domain.

2001 A survey of nearisometries, A volume dedicated to Olli Martio on the occasion of his 60th birthday, Report. Univ. Jyväskylä 83, 2001, 305-315. ps

2001 Isometries of normed spaces (together with T. Figiel and P. Semrl), Colloq. Math. 92, 2002, 153-154. pdf

Abstract: We relax the surjectivity condition of the Mazur-Ulam theorem.

2001 Isometric approximation property of unbounded sets, to appear in Results Math. pdf

Abstract: The problem of the paper below is solved for unbounded sets. A simple proof of the original Hyers-Ulam theorem is given.

2000 Isometric approximation property in euclidean spaces, Israel J. Math. 128, 2002, 1-27. ps

Abstract: A geometric characterization for bounded subsets A of a euclidean space E is given for the property that each s-nearisometry of A into E can be approximated by an isometry with error at most cs.

2000 Hyers-Ulam constants of Hilbert spaces (together with T. Huuskonen), Studia Math. 153, 2002, 31-40. pdf

Abstract: We show that for each s > 0 and for each Hilbert space E there is an s-nearisometry f of E onto E such that the sup distance d(T,f) for each isometry T is at least J(E)s, where J(E) is the Jung constant of E.

2000 Isometric approximation (together with P. Alestalo and D.A. Trotsenko), Israel J. Math 125, 2001, 61-82. ps

Abstract: We study isometric approximation of nearisometries and nearsymmetric maps of bounded sets of euclidean spaces.

1999 The free quasiworld, Banach Center Publ. 48, 1999, 55-118. pdf

Abstract: A mini monograph on freely quasiconformal and related maps in Banach spaces.

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