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Geometrical inequalities and mixed volumes in the local theory of Banach spaces. (English) Zbl 0582.46015

Colloq. Honneur L. Schwartz, Éc. Polytech. 1983, Vol. 1, Astérisque 131, 373-400 (1985).
[For the entire collection see Zbl 0566.00010.]
Finite dimensional normed spaces are studied in the paper using a geometrical approach. Search for Euclidean sections and projections on them is connected with mixed volumes of the unit ball of a space and some geometrical inequalities (Urysohn, Santalo and Alexandroff inequalities). A Levy mean approach is also used to estimate mixed volumes. Also necessary information on mixed volumes and Euclidean sections is included (as a survey part of the paper).
Note that most problems discussed in the paper are solved now: Problem 1, § 4 - in the positive [see V. Milman, Proc. Am. Math. Soc. 94, 445-449 (1985)]; Problem 2, § 4 and Problem 4.5 - in the positive [see V. Milman, C. R. Acad. Sci., Paris, Sér. I 302, 25-28 (1986)]; Problem 2, § 7 - in the negative (J. Bourgain-V. Milman; T. Figiel); Problem 1, § 7 is still open.

MSC:

46B20 Geometry and structure of normed linear spaces
26D20 Other analytical inequalities
46B25 Classical Banach spaces in the general theory

Citations:

Zbl 0566.00010