Uniqueness of Cartesian Products of Compact Convex Sets

• Autorzy: Lipecki Z. , Losert V. , Spurný J.

Identyfikatory

DOI

10.4064/ba59-2-7

• Języki publikacji: EN

Abstrakty

EN

Let X_i, i∈I, and Y_j, j∈J, be compact convex sets whose sets of extreme points are affinely independent and let φ be an affine homeomorphism of ∏_i∈IX_i onto ∏_j∈JY_j. We show that there exists a bijection b:I→J such that φ is the product of affine homeomorphisms of X_i onto Y_b(i), i∈I.

Słowa kluczowe

EN

compact convex set; simplex; extreme point; affine independence; affine homeomorphism; Cartesian product

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2011
• Tom: Vol. 59, no 2
• Strony: 175--183
• Opis fizyczny: Bibliogr. 23 poz.

Twórcy

autor

Lipecki Z.

• Institute of Mathematics Polish Academy of Sciences Wrocław Branch Kopernika 18, 51-617 Wrocław, Poland

autor

Losert V.

• Institut für Mathematik Universität Wien, 1090 Wien, Austria

autor

Spurný J.

• Department of Mathematical Analysis Faculty of Mathematics and Physics Charles University Sokolovská 83, 186 75 Praha 8, Czech Republic

Bibliografia

• [1] E. M. Alfsen, Compact Convex Sets and Boundary Integrals, Ergeb. Math. Grenzgeb. 57, Springer, New York, 1971.
• [2] L. Asimow and A. J. Ellis, Convexity Theory and its Applications in Functional Analysis, London Math. Soc. Monogr. 16, Academic Press, London, 1980.
• [3] B. Banaschewski and R. Lowen, A cancellation law for partially ordered sets and T0 spaces, Proc. Amer. Math. Soc. 132 (2004), 3463–3466.
• [4] E. Behrends and J. Pelant, The cancellation law for compact Hausdorff spaces and vector-valued Banach–Stone theorems, Arch. Math. (Basel) 64 (1995), 341–343.
• [5] K. Borsuk, Sur la décomposition des polyèdres en produits cartésiens, Fund. Math. 31 (1938), 137–148.
• [6] —, On the decomposition of a locally connected compactum into Cartesian product of a curve and a manifold, ibid. 40 (1953), 140–159.
• [7] R. Cauty, Sur les homéomorphismes de certains produits de courbes, Bull. Acad. Polon. Sci. Sér. Sci. Math. 27 (1979), 413–416.
• [8] G. Choquet, Lectures on Analysis. Vol. I–III: Infinite Dimensional Measures and Problem Solutions, W. A. Benjamin, New York, 1969.
• [9] A. J. Ellis and W. S. So, Isometries and the complex state spaces of uniform algebras, Math. Z. 195 (1987), 119–125.
• [10] V. P. Fonf, J. Lindenstrauss, and R. R. Phelps, Infinite dimensional convexity, in: Handbook of the Geometry of Banach Spaces, Vol. I, North-Holland, Amsterdam, 2001, 599–670.
• [11] S. Goldberg and A. H. Kruse, The existence of compact linear maps between Banach spaces, Proc. Amer. Math. Soc. 13 (1962), 808–811.
• [12] W. T. Gowers, A solution to the Schroeder–Bernstein problem for Banach spaces, Bull. London Math. Soc. 28 (1996), 297–304.
• [13] D. Handelman, Representing polynomials by positive linear functions on compact convex polyhedra, Pacific J. Math. 132 (1988), 35–62.
• [14] Z. Jelonek, On the cancellation problem, Math. Ann. 344 (2009), 769–778.
• [15] H. E. Lacey, The Isometric Theory of Classical Banach Spaces, Grundlehren Math. Wiss. 208, Springer, New York, 1974.
• [16] Z. Lipecki, Quasi-measures with finitely or countably many extreme extensions, Manuscripta Math. 97 (1998), 469–481.
• [17] —, On compactness and extreme points of some sets of quasi-measures and measures. III, ibid. 117 (2005), 463–473.
• [18] —, On compactness and extreme points of some sets of quasi-measures and measures. IV, ibid. 123 (2007), 133–146.
• [19] R. R. Phelps, Lectures on Choquet’s Theorem, 2nd ed., Lecture Notes in Math. 1757, Springer, Berlin, 2001.
• [20] Z. Semadeni, Banach Spaces of Continuous Functions. Vol. I, Monografie Mat. 55, PWN–Polish Sci. Publ., Warszawa, 1971.
• [21] V. Trnková, Homeomorphisms of products of subsets of the Cantor discontinuum, Dissertationes Math. (Rozprawy Mat.) 268 (1988).
• [22] S. M. Ulam, A Collection of Mathematical Problems, Interscience Tracts Pure Appl. Math. 8, Interscience, New York, 1960.
• [23] G. M. Ziegler, Lectures on Polytopes, Grad. Texts in Math. 152, Springer, New York, 1998.