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The dimension of Euclidean subspaces of quasi-normed spaces

Published online by Cambridge University Press:  24 October 2008

S. J. Dilworth
S. J. Dilworth
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge
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Extract

The purpose of this article is to extend certain results which are known to hold for convex bodies to a class of non-convex bodies occurring in the theory of topological vector spaces. In the first section after this introduction an analogue of F. John's Theorem on the distance of a finite dimensional space from Euclidean space is obtained, and the result is shown to be best possible. The Dvoretzky-Rogers Lemma on the points of contact of a symmetric convex body with the ellipsoid of maximum volume contained within it is discussed for certain non-convex bodies. In the next part the Dvoretzky Theorem on the existence of ellipsoidal sections is shown to hold with the best possible estimate for the dimension of the sections. It follows from estimates involving cotype constants that the finite dimensional subspaces of Lp (0 < p < 1) possess large almost Hilbertian subspaces. The final section extends the theorem of S. Szarek relating the volume of a body to the existence of ellipsoidal sections.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 97 , Issue 2 , March 1985 , pp. 311 - 320
Copyright
Copyright © Cambridge Philosophical Society 1985

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References

REFERENCES

[1]Dilworth, S. J.. On the dimension of almost Hilbertian subspaces of quotient spaces. J. London Math. Soc. (to appear).Google Scholar
[2]Dvoretzky, A.. Some results on convex bodies and Banach spaces. Proc. Int. Symp. on Linear Spaces, Jerusalem, 1961, 123–160.Google Scholar
[3]Dvoretzky, A. and Rogers, C.. Absolute and unconditional convergence in normed linear spaces. Proc. Nat. Acad. Sci. U.S.A. 36 (1950), 192–197.Google Scholar
[4]Eggleston, H. G.. Convexity (Cambridge University Press, 1958).Google Scholar
[5]Figiel, T. A.. A short proof of Dvoretzky's Theorem on almost spherical sections. Comp. Math. 33 (1976), 297–301.Google Scholar
[6]Figiel, T. A., Lindenstrauss, J. and Milman, V. D.. The dimension of almost spherical sections of convex bodies. Acta Math. 139 (1977), 53–94.Google Scholar
[7]John, F.. Extremum Problems with Inequalities as Subsidiary Conditions. Courant anniversary volume (Interscience, New York, 1948), 187–204.Google Scholar
[8]Köthe, G.. Topological Vector Spaces I (Springer, 1969).Google Scholar
[9]Krivine, J. L.. Sous espaces de dimension fini des espaces de Banach reticulés. Ann. Math. 104 (1976), 1–29.Google Scholar
[10]Gurarii, V. I., Kadec, M. I. and Macaev, V. I.. On distances between finite dimensional analogues of Lp spaces. Mat. Sb. 70 (1966), 481–489.Google Scholar
[11]Milman, V. D.. A new proof of the theorem of A. Dvoretzky on sections of convex bodies. Funct. Anal. Appl. 5 (1971), 28–37 (translated from Russian).Google Scholar
[12]Rogalski, M.. Sur le quotient volumique d'un espace de dimension fini (Seminaire Beauzamy-Maurey-Krivine, 1981).Google Scholar
[13]Rolewicz, S.. On a certain class of linear metric spaces. Bull. Acad. Polon. Sci. C. III 5 (471–473), 1957.Google Scholar
[14]Santalo, L.. Un invariante affin para los cuerpos del espacio de n dimensions. Portugal. Math. 8 (1949), 155–161.Google Scholar
[15]Schwartz, L.. Geometry and Probability in Banach Spaces (Springer, 1981).Google Scholar
[16]Szankowski, A.. On Dvoretzky's Theorem on almost spherical sections of convex bodies. Israel J. Math. 17 (1974), 326–338.Google Scholar
[17]Szarek, S.. On Kashin almost euclidean orthogonal decomposition of . Bull Acad. Polon. Sci. 26 (1978), 691–694.Google Scholar
[18]Szarek, S.. Volume estimates and nearly euclidean decompositions for normed spaces. Séminaire d' Analyse Fonct., no. 25. (Ecole Polytechnique, 1979–1980).Google Scholar
[19]Szarek, S. and Tomczak-Jaegermann, N.. On nearly euclidean decompositions for some classes of Banach spaces. Comp. Math. 40 (1980), 367–385.Google Scholar