Milman, V. D. 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. Cited in 1 ReviewCited in 9 Documents MSC: 46B20 Geometry and structure of normed linear spaces 26D20 Other analytical inequalities 46B25 Classical Banach spaces in the general theory Keywords:local theory; Finite dimensional normed spaces; Euclidean sections and projections on them; mixed volumes of the unit ball; geometrical inequalities; Urysohn, Santalo and Alexandroff inequalities; Levy mean approach Citations:Zbl 0566.00010 PDFBibTeX XML