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On the topological classification of starlike bodies in Banach spaces

Azagra Rueda, Daniel and Dobrowolski, Tadeusz (2003) On the topological classification of starlike bodies in Banach spaces. Topology and its Applications, 132 (3). pp. 221-234. ISSN 0166-8641

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Abstract

Starlike bodies are interesting in nonlinear analysis because they are strongly related to polynomials and smooth bump functions, and their topological and geometrical properties are therefore worth studying. In this note we consider the question as to what extent the known results on topological classification of convex bodies can be generalized for the class of starlike bodies, and we obtain two main results in this line, one which follows the traditional Bessaga-Klee scheme for the classification of convex bodies (and which in this new setting happens to be valid only for starlike bodies whose characteristic cones are convex), and another one which uses a new classification scheme in terms of the homotopy type of the boundaries of the starlike bodies (and which holds in full generality provided the Banach space is infinite-dimensional).


Item Type:Article
Uncontrolled Keywords:Convex bodies; Manifolds; Smooth; Homeomorphisms; Negligibility; Spines; Starlike body; Homeomorphism
Subjects:Sciences > Mathematics > Mathematical analysis
ID Code:14502
References:

[1] F.D. Ancel, C.R. Guilbault, Campact contractible n-manifolds have are of spines (n > 5), Pacific J. Malb. 168 (1995) 1-10.

[2] R.D. Anderson, J.D. MeCharen, On extending homeomorphisms to Fréchet manifolds, Prac. Amer. Math. Soc. 25 (1970) 283-289.

[3] D. Azagra, M. Cepedello, Smooth Lipschitz retractions ofstarlike bodies anto their boundaries in infinitedimensional Banach spaces, Bull. London Math. Soco 33 (2001) 443--453.

[4] D. Azagra, R. Deville, James' theorem fails for starlike bodies, J. Funct. Anal. 180 (2001) 328-346.

[5] D. Azagra, T. Dobrowolski, Smooth negligibility of compact sets in infinite-dimensional Banach spaces, with applications, Math. Ann. 312 (3) (1998) 445-463.

[6] C. Bessaga, VL. Klee, Two topological properties of topological linear spaces, Israel J. Math. 2 (1964) 211-220.

[7] C. Bessaga, VL. Klee, Every non-normable Fréchet space is homeomorphic with all of its cl0sed convex bodies, Math. Ann. 163 (1966) 161-166.

[8] C. Bessaga, A. Pelczynski, Selected Topics in Infinite-Dimensional Topology, in: Monografie Matematyczne, Warszw.va, 1975.

[9] D. Burghe1ea, N.H. Kuiper, Hilbert manifolds, Ann. of Math. 90 (1969) 379-417.

[10] H.H. Corson, VL. Klee, Topological c1assification of convex sets, in: Proc. Symp. Pure Math., Vol. 7. Convexity, American Mathematical Society, Providence, RI, 1963, pp. 37-51.

[11] M.L. Curtis, K.W. Kwun, Infinite sums ofmanifolds, Topology 6 (1965) 31-42.

[12] T. Dobrowolski, Smooth and R-analytic negligibility ofsubsets and extension ofhomeomorphism in Banach spaces, Studia Math. 65 (1979) 115-139.

[13] T. Dobrowolski, Relative c1assification ofsmooth convex bodies, Bull. Acad. Poloo. Sci. Sér. Sci. Math. 25 (1977) 309-312.

[14] J. Eells, D. Elworthy, Open embeddings of certain Banach manifolds, Ano. of Math. 91 (1970) 465--485.

[15] D. Elworthy, Embeddings, isotopy and stability of Banach manifolds, Compositio Math. 24 (1972) 175-226.

[16] L.C Glaser, Uncountably many contractible open 4-manifolds, Topology 6 (1967) 37--42.

[17] C.R. Guilbault, Sorne campact contractible manifold containing disjoint spines, Topology 34 (1995) 99-108.

[18] VL. Klee, Convex bodies and periodic homeomorphisms in Hilbert space, Trans. Amer. Math. Soco 74 (1953) 10-43.

[19] D.R. McMillan, Sorne contractible open 3-manifolds, Trans. Amer. Math. Soco 102 (1962) 373-382.

[20] N. Moulis, Sur les variétés hilbertiennes et les fonctions non dégénérées, Indag. Math. 30 (1968) 497-511.

[21] T.B. Rushing, Topological Embeddings, Academic Press, Ne\V York, 1973.

[22] lJ. Stoker, Unbounded convex point sets, Amer. J. Math. 62 (1940) 165-179.

Deposited On:01 Feb 2012 16:07
Last Modified:01 Feb 2012 16:27

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