Auerbach Bases and Minimal Volume Sufficient Enlargements

Ostrovskii, M. I.

Canadian mathematical bulletin, Tome 54 (2011) no. 4, pp. 726-738

Résumé

Let ${{B}_{Y}}$ denote the unit ball of a normed linear space $Y$ . A symmetric, bounded, closed, convex set $A$ in a finite dimensional normed linear space $X$ is called a sufficient enlargement for $X$ if, for an arbitrary isometric embedding of $X$ into a Banach space $Y$ , there exists a linear projection $P:\,Y\,\to \,X$ such that $P({{B}_{Y}})\,\subset \,A$ . Each finite dimensional normed space has a minimal-volume sufficient enlargement that is a parallelepiped; some spaces have “exotic” minimal-volume sufficient enlargements. The main result of the paper is a characterization of spaces having “exotic” minimal-volume sufficient enlargements in terms of Auerbach bases.

Détail

DOI : 10.4153/CMB-2011-043-3

Mots-clés : 46B07, 52A21, 46B15

Bibliographie

[1] [1] Auerbach, H., On the area of convex curves with conjugate diameters (in Polish). Ph. D. thesis, University of Lwów, 1930. Google Scholar

[2] [2] Banach, S., Théorie des opérations linéaires, Monografje Matematyczne I, Warszawa, 1932. Google Scholar

[3] [3] Day, M. M., Polygons circumscribed about closed convex curves.. Trans. Amer. Math. Soc. 62(1947), 315–319. Google Scholar

[4] [4] Erdahl, R. M., Zonotopes, dicings, and Voronoi's conjecture on parallelohedra. European J. Combin. 20(1999), no. 6, 527–549. doi:10.1006/eujc.1999.0294 Google Scholar

[5] [5] Gordon, Y., Meyer, M., and Pajor, A., Ratios of volumes and factorization through ℓ . Illinois J. Math. 40(1996), no. 1, 91–107. Google Scholar

[6] [6] Johnson, W. B. and Lindenstrauss, J., Basic concepts in the geometry of Banach spaces. In: Handbook of the geometry of Banach spaces, 1, North-Holland, Amsterdam, 2001, pp. 1–84. Google Scholar

[7] [7] McMullen, P., Space tiling zonotopes. Mathematika 22(1975), no. 2, 202–211. doi:10.1112/S0025579300006082 Google Scholar

[8] [8] Orrick, W. P. and Solomon, B., Large-determinant sign matrices of order 4k + 1 . Discrete Math. 307(2007), no. 2, 226–236. doi:10.1016/j.disc.2006.04.041 Google Scholar

[9] [9] Ostrovskii, M. I., Generalization of projection constants: sufficient enlargements. Extracta Math. 11(1996), no. 3, 466–474. Google Scholar

[10] [10] Ostrovskii, M. I., Projections in normed linear spaces and sufficient enlargements. Arch. Math. (Basel) 71(1998), no. 4, 315–324. Google Scholar

[11] [11] Ostrovskii, M. I., Minimal-volume projections of cubes and totally unimodular matrices.. Linear Algebra Appl. 364(2003), 91–103. doi:10.1016/S0024-3795(02)00539-6 Google Scholar

[12] [12] Ostrovskii, M. I., Sufficient enlargements of minimal volume for two-dimensional normed spaces. Math. Proc. Cambridge Philos. Soc. 137(2004), no. 2, 377–396. doi:10.1017/S0305004104007819 Google Scholar

[13] [13] Ostrovskii, M. I., Sufficient enlargements of minimal volume for finite-dimensional normed linear spaces. J. Funct. Anal. 255(2008), no. 3, 589–619. doi:10.1016/j.jfa.2008.04.012 Google Scholar

[14] [14] Ostrovskii, M. I., Sufficient enlargements in the study of projections in normed linear spaces. Indian J. Math. 2008, suppl, 105–122. Google Scholar

[15] [15] Pełczyński, A. and Szarek, S. J., On parallelepipeds of minimal volume containing a convex symmetric body in ℝ n . Math. Proc. Cambridge Phil. Soc. 109(1991), no. 1, 125–148. doi:10.1017/S0305004100069619 Google Scholar

[16] [16] Plichko, A. M., On the volume method in the study of Auerbach bases of finite-dimensional normed spaces. Colloq. Math. 69(1995), no. 2, 267–270. Google Scholar

[17] [17] Schneider, R., Convex Bodies: the Brunn–Minkowski theory. In: Encyclopedia of Mathematics and its Applications, 44, Cambridge University Press, Cambridge, 1993. Google Scholar

[18] [18] Taylor, A. E., A geometric theorem and its application to biorthogonal systems.. Bull. Amer. Math. Soc. 53(1947), 614–616. doi:10.1090/S0002-9904-1947-08855-8 Google Scholar