Abstract
We provide a generalization of John's representation of the identity for the maximal volume position of L inside K, where K and L are arbitrary smooth convex bodies in ℝn. From this representation we obtain Banach–Mazur distance and volume ratio estimates.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
[Ba] Ball, K. M.: Volume ratios and a reverse isoperimetric inequality, J. London Math. Soc. (2) 44 (1991), 351-359.
[Bar] Barthe, F.: Inégalités de Brascamp-Lieb et convexité, C.R. Acad. Sci. Paris 324 (1997), 885-888.
[Be] Beer G.: Approximate selections for upper semicontinuous convex valued multi-functions, J. Approx. Theory 39 (1983), 172-184.
[BL] Brascamp, H. and Lieb, E. H.: Best constants in Young's inequality, its converse and its generalization to more than three functions, Adv. in Math. 20 (1976), 151-173.
[DR] Dvoretzky, A. and Rogers, C. A.: Absolute and unconditional convergence in normed linear spaces, Proc. Nat. Acad. Sci. USA 36 (1950), 192-197.
[Gl] Gluskin, E. D.: The diameter of the Minkowski compactum is approximately equal to n, Funct. Anal. Appl. 15 (1981), 72-73.
[Gr] Grünbaum, B.: Measures of symmetry for convex sets, In: Convexity VII, Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, 1963, pp. 233-270.
[GM] Giannopoulos, A. A. and Milman, V. D.: Extremal problems and isotropic positions of convex bodies, Israel J. Math. 117 (2000), 29-60.
[J] John, F.: Extremum Problems with Inequalities as Subsidiary Conditions, Courant Anniversary Volume, New York, 1948, pp. 187-204.
[L] Lassak, M.: Approximation of convex bodies by centrally symmetric bodies, Geom. Dedicata 72 (1998), 63-68.
[LTJ] Litvak, A. and Tomczak-Jaegermann, N.: Random aspects of high-dimensional con-vex bodies, In: Milman and Schechtman (eds), Lecture Notes in Math. 1745, Springer, New York, 2000, pp. 168-190.
[P] Palmon, O.: The only convex body with extremal distance from the ball is the simplex, Israel J. Math. 80 (1992), 337-349.
[R] Rudelson, M.: Distances between non-symmetric convex bodies and the MM*-estimate, Positivity 4 (2000), 161-178.
[RW] Rockafellar, R. T. and Wets, R. J-B.: Variational Analysis, Grundl. Math. Wissen. 317, Springer, Berlin, 1998.
[TJ] Tomczak-Jaegermann, N.: Banach-Mazur Distances and Finite Dimensional Operator Ideals, Pitman Monographs 38, Pitman, London, 1989.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Giannopoulos, A., Perissinaki, I. & Tsolomitis, A. John's Theorem for an Arbitrary Pair of Convex Bodies. Geometriae Dedicata 84, 63–79 (2001). https://doi.org/10.1023/A:1010327006555
Issue Date:
DOI: https://doi.org/10.1023/A:1010327006555