References
[A] Alexandrov, A.: On the theory of mixed volumes of convex bodies. I. Math. Sbornik2 (N5), 947–972 (1937); II. idem Math. Sbornik2 (N6) 1205–1238; III. idem Math. Sbornik3 (NI) 27–46 (1938);
[Ba] Bambah., R.P.: Polar reciprocal convex bodies. Proc. Camb. Philos. Soc.51, 377–378 (1954)
[Be] Bernstein, D.N.: The number of roots of a system of equations. Funkts. Anal. Prilozh.9 (N3), 1–4 (1975)
[Bo, 1] Bourgain, J.: On Martingale transforms in finite dimensional lattices with an appendix on theK-convexity constant. Math. Nachr.119, 41–53 (1984)
[Bo, 2] Bourgain, J.: A remark on entropy on Abelian, groups and the uniform approximation property. Studia Math. (in press) (1987)
[B-M] Bourgain, J., Milman, V.D.: Sections euclidiennes et volume des corps symétriques convexes dans ℝn. C.R. Acad. Sci., Paris, Sér.I, t 300,13, 435–438 (1985)
[B-Z] Burago, Y.D., Zalgaler, V.A.: Geometrical inequalities. Leningrad: Nauka 1980
[D-M-T] Davis, W.J., Milman, V.D., Tomczak-Jaegermann, N.: The distance between certainn-dimensional Banach spaces. Israel J. Math.39, 1–15 (1982)
[Fe] Fenchel, W.: Inégalités quadratiques entre les volumes mixtes des corps convexes. C.R. Acad. Sci., Paris, Ser. I.,203, 647–650 (1936)
[F] Firey, W.M.J.: Support flags to convex bodies, Geom. Dedic2, 225–248 (1973)
[F-L-M] Figiel, T., Lindenstrauss, J., Milman, V.D.: The dimension of almost spherical sections of convex bodies. Acta Math.139, 53–94 (1977)
[F-T] Figiel, T., Tomczak-Jaegermann, N.: Projections onto Hilbertian subspaces of Banach spaces. Israel J. Math.53, 155–171 (1979)
[G-R] Gordon, Y., Reisner, S.: Volume estimates and applications to finite-dimensional Banach spaces. Preprint
[J] John, F.: Extremum problems with inequalities as subsidiary conditions. Courant anniversary volume, pp. 187–204. New York: Interscience 1948
[K] Kauchnirenko, A.G.: Polyèdres de Newton et nombres de Milnor. Invent. Math.32, 1–31 (1979)
[L] Lekkerkerker, C.G.: Geometry of numbers. Amsterdam: North-Holland 1969
[Lew] Lewis, D.R.: Ellipsoides defined by Banach ideal norms. Mathematica26, 18–29 (1979)
[Lu] Lutwak, E.: On cross-sectional measures of polar reciprocal convex bodies. Geom. Dedicata1, 79–80 (1976)
[Ma] Mahler, K.: Ein Übertragungsprincip für konvexe Körper. Casopis Pest. Mat. Fys.68, 93–102 (1909)
[Ma-P] Maurey, B., Pisier, G.: Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach. Stud. Math.58, 45–90 (1976)
[Mi, 1] Milman, V.D.: Almost Euclidean quotient spaces of subspaces of finite dimensional normed spaces. Proc. Am. Math. Soc.94, 445–449 (1985)
[Mi, 2] Milman, V.D.: Random subspaces of proportional dimension of finite dimensional normed spaces; approach through isoperimetric inequality. Missouri Conf. 1984. Springer LNM 1166, 1985, pp. 106–116
[Mi, 3] Milman, V.D.: Volume approach and iteration procedures in local theory of normed spaces. Missouri Conf. 1984, Springer LNM 1166, 1985, pp. 99–105
[Mi,4] Milman, V.D.: Inégalité de Brunn-Minkowski inverse et applications à la théorie locale des espaces normés. C.R. Acad. Sci., Paris, Sér. I.,7302, 25–28 (1986)
[M-S] Milman, V.D., Schechtman, G.: Asymptotic theory of finite dimensional normed spaces. (Lect. Notes in Math., vol. 1200) Berlin-Heidelberg-New York: Springer 1986
[P-T] Pajor, A., Tomczak-Jaegerman, N.: Nombres de Gelfand et sections euclidiennes de grande dimension. Séminaire d'Analyse fonctionnelle, 1984/85, Univ. Paris VI et VII
[Pel] Pelczynski, A.: Geometry of infinite dimensional Banach spaces and operator ideals. Notes in Banach spaces, pp. 81–181. Austin and London: Univ. of Texas Press 1980
[Pi, 1] Pisier, G.:K-convexity, Proceeding of Research Workshop on Banach Spaces, University of Iowa, 1981, pp. 139–151 Algo: Remarque sur un résultat non-publié de B. Maurey, Sém. d'Analyse Fonctionnelle, 1980/81, Exp. v., Ecole Polytechnique
[Pi, 2] Pisier, G.: Private communications
[R] Reisner, S.: Random polytopes and volume product of symmetric bodies. Math. Scand.57, 386–392 (1985)
[Ro] Rogalski, M.: Sur le quotient volumique d'un espace de dimension fini. Publ. Math. Univ. Pierre et Marie Curie46, C3 (1980/81)
[R-S] Rogers, C.A., Shephard, G.C.: The difference body of a convex body. Arch. Math.8, 220–233 (1957)
[S, 1] Saint-Raymond, J.: Sur le volume des corps convexes symétriques. Séminaire d'initiation à l'analyse 1980/81, Paris VI University
[S, 2] Saint-Raymond, J.: Le volume des idéaux d'opératurs classiques. Stud. Math.80, 63–75 (1984)
[Sa, 1] Santalo, L.A.: Un invariante afin pasa los cuerpos convexos del espacio den-dimensiones. Port. Math.8, 155–161 (1949)
[Sa, 2] Santalo, L.A.: Integral geometry and geometric probability. Encyclopedia Math. and Appl., Reading, MA: Addison-Wesley 1976
[Sz] Szarek, S.: On Kashin's almost Euclidean orthogonal decomposition ofl 1 n . Bull. Acad. Pol. Sci., Ser. Sci. Math.26, 691–694 (1978)
[Sz-T] Szarek, S., Tomczak-Jaegermann, N.: On nearly Euclidean decompositions for some classes of Banach spaces. Compos. Math.40, 367–385 (1980)
[Zy] Zygmund, A.: Trigonometric series, Cambridge UP, 1959
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bourgain, J., Milman, V.D. New volume ratio properties for convex symmetric bodies in ℝn . Invent Math 88, 319–340 (1987). https://doi.org/10.1007/BF01388911
Issue Date:
DOI: https://doi.org/10.1007/BF01388911