Book Review
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MathSciNet review:
1567936
Full text of review:
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Book Information:
Author:
Gilles Pisier
Title:
The volume of convex bodies and Banach space geometry
Additional book information:
Cambridge Tracts in Mathematics, vol. 94, Cambridge University Press, Cambridge, 1989, xv + 250 pp., $49.50. ISBN 0-521-364655.
S. Banach, Theory of linear operations, North-Holland Mathematical Library, vol. 38, North-Holland Publishing Co., Amsterdam, 1987. Translated from the French by F. Jellett; With comments by A. Pełczyński and Cz. Bessaga. MR 880204
J. Bourgain and V. D. Milman, New volume ratio properties for convex symmetric bodies in $\textbf {R}^n$, Invent. Math. 88 (1987), no. 2, 319–340. MR 880954, DOI 10.1007/BF01388911
Aryeh Dvoretzky, Some results on convex bodies and Banach spaces, Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960) Jerusalem Academic Press, Jerusalem; Pergamon, Oxford, 1961, pp. 123–160. MR 0139079
T. Figiel, J. Lindenstrauss, and V. D. Milman, The dimension of almost spherical sections of convex bodies, Acta Math. 139 (1977), no. 1-2, 53–94. MR 445274, DOI 10.1007/BF02392234
Yehoram Gordon, Some inequalities for Gaussian processes and applications, Israel J. Math. 50 (1985), no. 4, 265–289. MR 800188, DOI 10.1007/BF02759761
Yehoram Gordon, Gaussian processes and almost spherical sections of convex bodies, Ann. Probab. 16 (1988), no. 1, 180–188. MR 920263
Fritz John, Extremum problems with inequalities as subsidiary conditions, Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, Interscience Publishers, Inc., New York, N. Y., 1948, pp. 187–204. MR 0030135
W. B. Johnson, H. König, B. Maurey, and J. R. Retherford, Eigenvalues of $p$-summing and $l_{p}$-type operators in Banach spaces, J. Functional Analysis 32 (1979), no. 3, 353–380. MR 538861, DOI 10.1016/0022-1236(79)90046-6
Jean-Pierre Kahane, Une inégalité du type de Slepian et Gordon sur les processus gaussiens, Israel J. Math. 55 (1986), no. 1, 109–110 (French, with English summary). MR 858463, DOI 10.1007/BF02772698
B. Knaster, Problème 4, Colloq. Math. 1 (1947), 30-31.
Kurt Mahler, Ein Übertragungsprinzip für konvexe Körper, Časopis Pěst. Mat. Fys. 68 (1939), 93–102 (German). MR 0001242
Bernard Maurey and Gilles Pisier, Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach, Studia Math. 58 (1976), no. 1, 45–90 (French). MR 443015, DOI 10.4064/sm-58-1-45-90
Mathieu Meyer and Alain Pajor, On Santaló’s inequality, Geometric aspects of functional analysis (1987–88), Lecture Notes in Math., vol. 1376, Springer, Berlin, 1989, pp. 261–263. MR 1008727, DOI 10.1007/BFb0090059
V. D. Milman, A new proof of A. Dvoretzky’s theorem on cross-sections of convex bodies, Funkcional. Anal. i Priložen. 5 (1971), no. 4, 28–37 (Russian). MR 0293374
V. D. Milman, Almost Euclidean quotient spaces of subspaces of a finite-dimensional normed space, Proc. Amer. Math. Soc. 94 (1985), no. 3, 445–449. MR 787891, DOI 10.1090/S0002-9939-1985-0787891-1
Vitali D. Milman, Inégalité de Brunn-Minkowski inverse et applications à la théorie locale des espaces normés, C. R. Acad. Sci. Paris Sér. I Math. 302 (1986), no. 1, 25–28 (English, with French summary). MR 827101
V. D. Milman, A few observations on the connections between local theory and some other fields, Geometric aspects of functional analysis (1986/87), Lecture Notes in Math., vol. 1317, Springer, Berlin, 1988, pp. 283–289. MR 950988, DOI 10.1007/BFb0081748
Gilles Pisier, Holomorphic semigroups and the geometry of Banach spaces, Ann. of Math. (2) 115 (1982), no. 2, 375–392. MR 647811, DOI 10.2307/1971396
Gilles Pisier, Weak Hilbert spaces, Proc. London Math. Soc. (3) 56 (1988), no. 3, 547–579. MR 931514, DOI 10.1112/plms/s3-56.3.547
J. Saint-Raymond, Sur le volume des corps convexes symétriques, Initiation Seminar on Analysis: G. Choquet-M. Rogalski-J. Saint-Raymond, 20th Year: 1980/1981, Publ. Math. Univ. Pierre et Marie Curie, vol. 46, Univ. Paris VI, Paris, 1981, pp. Exp. No. 11, 25 (French). MR 670798
L. Santaló, Un invariante afin para los cuerpos convexos del espacio de n dimensions, Portugal. Math. 8 (1949), 155-161.
Review Information:
Reviewer:
Donald L. Burkholder
Journal:
Bull. Amer. Math. Soc. 25 (1991), 140-145
DOI:
https://doi.org/10.1090/S0273-0979-1991-16046-5