Bulletin of the American Mathematical Society

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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.

S. Banach, Theory of linear operations, with a supplement by A. Pełczyński and C. Bessaga on Some aspects of the present theory of Banach spaces, North-Holland, Amsterdam, 1987. MR 0880204




J. Bourgain and V. D. Milman, New volume ratio properties for convex symmetric bodies in R, Invent. Math. 88 (1987), 319-340. MR 880954




A. Dvoretzky, Some results on convex bodies and Banach spaces, Proceedings of the International Symposium on Linear Spaces held at the Hebrew University of Jerusalem, July 5-12, 1960, The Israel Academy of Sciences and Humanities, Jerusalem, 1961, pp. 123-160. MR 139079




T. Figiel, J. Lindenstrauss, and V. D. Milman, The dimension of almost spherical sections of convex bodies, Acta Math. 139 (1977), 53-94. MR 445274




Y. Gordon, Some inequalities for Gaussian processes and applications, Israel J. Math. 50 (1985), 265-289. MR 800188




Y. Gordon, Gaussian processes and almost spherical sections of convex bodies, Ann. Probab. 16 (1988), 180-188. MR 920263




F. John, Extremum problems with inequalities as subsidiary conditions, Studies and Essays presented to R. Courant on his 60th birthday, January 8, 1948, Interscience, New York, 1948, pp. 187-204. MR 30135




W. B. Johnson, H. König, B. Maurey, and J. R. Retherford, Eigenvalues of psumming and l, J. Funct. Anal. 32 (1979), 353-380. MR 538861




J. P. Kahane, Une inégalité du type de Slepian et Gordon sur les processus Gaussiens, Israel J. Math. 55 (1986), 109-110. MR 858463




B. Knaster, Problème 4, Colloq. Math. 1 (1947), 30-31.




K. Mahler, Ein Übertragungsprinzip für konvexe Körper, Časopis Pěst. Mat. 68 (1939), 93-102. MR 1242




B. Maurey and G. Pisier, Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach, Studia Math. 58 (1976), 45-90. MR 443015




M. Meyer and A. Pajor, On Santaló's inequality, GAFA Seminar, 1987-88, Lecture Notes in Math., vol. 1376, Springer-Verlag, Berlin and New York, 1989, pp. 261-263. MR 1008727




V. D. Milman, New proof of the theorem of Dvoretzky on sections of convex bodies, Funkcional Anal. i Prilozen 5 (1971), 28-37. (Russian) MR 293374




V. D. Milman, Almost Euclidean quotient spaces of subspaces of a finite-dimensional normed space, Proc. Amer. Math. Soc. 94 (1985), 445-449. MR 787891




V. D. Milman, Inégalité de Brunn-Minkowski inverse et applications à la théorie locale des espaces normés, C. R. Acad. Sci. Paris 302 (1986), 25-28. MR 827101




V. D. Milman, On a few observations on the connections between local theory and some other fields, GAFA Seminar, 1986-87, Lecture Notes in Math., vol. 1317, Springer-Verlag, Berlin and New York, 1988, pp. 283-289. MR 950988




G. Pisier, Holomorphic semi-groups and the geometry of Banach spaces, Ann. of Math. (2) 115 (1982), 375-392. MR 647811




G. Pisier, Weak Hilbert spaces, Proc. London Math. Soc. 56 (1988), 547-579. MR 931514




J. Saint-Raymond, Sur le volume des corps convexes symétriques, Séminaire Initiation à l'Analyse, 1980-81, Université P. et M. Curie, Paris, 1981. MR 670798




L. Santaló, Un invariante afin para los cuerpos convexos del espacio de n dimensions, Portugal. Math. 8 (1949), 155-161.