Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-20T06:23:59.783Z Has data issue: false hasContentIssue false
  • English
  • Français

Infinite-Dimensional Polyhedrality

Published online by Cambridge University Press:  20 November 2018

Vladimir P. Fonf  and
Libor Veselý
Vladimir P. Fonf
Affiliation:
Department of Mathematics, Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva 84105, Israel e-mail: [email protected]
Libor Veselý
Affiliation:
Dipartimento di Matematica, Università degli Studi, Via C. Saldini 50, 20133 Milano, Italy e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper deals with generalizations of the notion of a polytope to infinite dimensions. The most general definition is the following: a bounded closed convex subset of a Banach space is called a polytope if each of its finite-dimensional affine sections is a (standard) polytope.

We study the relationships between eight known definitions of infinite-dimensional polyhedrality. We provide a complete isometric classification of them, which gives solutions to several open problems. An almost complete isomorphic classification is given as well (only one implication remains open).

Keywords

46B2046B0346B0452B99
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 56 , Issue 3 , 01 June 2004 , pp. 472 - 494
Copyright
Copyright © Canadian Mathematical Society 2004

References

[1] Amir, D. and Deutsch, F., Suns, moons, and quasi-polyhedra. J. Approx. Theory 6(1972), 176–201.Google Scholar
[2] Bessaga, C. and Pełczyński, A., Spaces of continuous functions IV. On isomorphical classification of spaces of continuous functions. Studia Math. 19(1960), 53–62.Google Scholar
[3] Brosowski, B. and Deutsch, F., On some geometric properties of suns. J. Approx. Theory 10(1974), 245–267.Google Scholar
[4] Durier, R. and Michelot, C., Sets of efficient points in a normed space. J. Math. Anal. Appl. 117(1986), 506–528.Google Scholar
[5] Durier, R. and Papini, P. L., Polyhedral norms and related properties in infinite-dimensional Banach spaces: a survey. Atti Sem. Mat. Fis. Univ.Modena 40(1992), 623–645.Google Scholar
[6] Durier, R. and Papini, P. L., Polyhedral norms in an infinite dimensional space. Rocky Mountain J. Math. 23(1993), 863–875.Google Scholar
[7] Fonf, V. P., A property of spaces of continuous functions on segments of ordinals. Sibirsk. Mat. Zh. 21(1980), 230–232, (Russian).Google Scholar
[8] Fonf, V. P., Polyhedral Banach spaces. Math. Notes 30(1981), 809–813.Google Scholar
[9] Fonf, V. P., A form of infinite-dimensional polytope. Funktsional. Anal. i Prilozhen. 18(1984), 79–80; English tranls. in Funct. Anal. Appl. 18(1984), 154–155.Google Scholar
[10] Fonf, V. P., Three characterizations of polyhedral Banach spaces. Ukrain. Mat. Zh. 42(1990), 1286–1290; English transl. in Ukrainian Math. J. 42(1990), 1145–1148.Google Scholar
[11] Fonf, V. P., On the boundary of a polyhedral Banach space. Extracta Math. 15(2000), 145–154.Google Scholar
[12] Fonf, V. P. and Lindenstrauss, J., Some results on infinite-dimensional convexity. Israel J. Math. 108(1998), 13–32.Google Scholar
[13] Fonf, V. P., Lindenstrauss, J. and Phelps, R. R., Infinite Dimensional Convexity. Handbook of the Geometry of Banach Spaces 1, (eds.,Johnson, W. B. and Lindenstrauss, J.), Elsevier Science, 2001, 599–670.Google Scholar
[14] Gleit, A. and McGuigan, R., A note on polyhedral Banach spaces. Proc. Amer.Math. Soc. 33(1972), 398–404.Google Scholar
[15] Kadets, M. I., private communication, 1981.Google Scholar
[16] Klee, V., Some characterizations of convex polyhedra. Acta Math. 102(1959), 79–107.Google Scholar
[17] Klee, V., Polyhedral sections of convex bodies. Acta Math. 103(1960), 243–267.Google Scholar
[18] Lindenstrauss, J. and Tzafriri, L., Classical Banach Spaces I, Sequence Spaces. Springer-Verlag, New York, 1977.Google Scholar
[19] Maserick, P. H., Convex polytopes in linear spaces. Illinois J. Math 9(1965), 623–635.Google Scholar
[20] Rudin, W., Continuous functions on compact spaces without perfect subsets. Proc. Amer. Math. Soc. 8(1957), 39–42.Google Scholar
[21] Veselý, L., Boundary of polyhedral spaces: an alternative proof. Extracta Math. 15(2000), 213–217.Google Scholar