Abstract
LetB be a convex body in ℝn and let ɛ be an ellipsoid of minimal volume containingB. By contact points ofB we mean the points of the intersection between the boundaries ofB and ɛ. By a result of P. Gruber, a generic convex body in ℝn has (n+3)·n/2 contact points. We prove that for every ɛ>0 and for every convex bodyB ⊂ ℝn there exists a convex bodyK having
contact points whose Banach-Mazur distance toB is less than 1+ɛ.
We prove also that for everyt>1 there exists a convex symmetric body Γ ⊂ ℝn so that every convex bodyD ⊂ ℝn whose Banach-Mazur distance to Γ is less thant has at least (1+c 0/t 2)·n contact points for some absolute constantc 0.
We apply these results to obtain new factorizations of Dvoretzky-Rogers type and to estimate the size of almost orthogonal submatrices of an orthogonal matrix.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
[B1] K. Ball,Volumes of sections of cubes and related problems, Lecture Notes in Mathematics1376, Springer, Berlin, 1989, pp. 251–260.
[B2] K. Ball,Volume ratios and a reverse isoperimetric inequality, Journal of the London Mathematical Society44 (1991), no. 2, 351–359.
[B-S] J. Bourgain and S. Szarek,The Banach-Mazur distance to the cube and the Dvoretzky-Rogers factorization, Israel Journal of Mathematics62 (1988), 169–180.
[Gi1] A. A. Giannopoulos,A note on the Banach-Mazur distance to the cube, Operator Theory Advances and Applications77 (1995), 67–73.
[Gi2] A. A. Giannopoulos,A proportional Dvoretzky-Rogers factorization result, Preprint.
[G1] E. Gluskin,The diameter of the Minkowski compactum is approximately equal to n, Functional Analysis and its Applications15 (1981), 72–73 (in Russian).
[G2] E. Gluskin,Finite dimensional analogs of spaces without basis, Doklady Adakemii Nauk SSSR216 (1981), 1046–1050 (in Russian).
[Gr] M. Gruber,Minimal ellipsoids and their duals, Rendiconti del Circolo Matematico di Palermo, Ser. 237 (1988), 35–64.
[J] F. John,Extremum problems with inequalities as subsidiary conditions, inCourant Anniversary Volume, Interscience, New York, 1948, pp. 187–204.
[K-T] B. Kashin and L. Tzafriri,Some remarks on the restrictions of operators to coordinate subspaces, Preprint.
[L-T] M. Ledoux and M. Talagrand,Probability in Banach spaces, Ergeb. Math. Grenzgeb., 3 Folge, vol. 23, Springer, Berlin, 1991.
[Pa-T-J] A. Pajor and N. Tomczak-Jaegermann,Subspaces of small codimension of finite dimensional Banach spaces, Proceedings of the American Mathematical Society97 (1986), 637–642.
[P-T-J] A. Pelczynski and N. Tomczak-Jaegermann,On the length of faithful nuclear representations of finite rank operators, Mathematika35 (1988), no. 1, 126–143.
[R] M. Rudelson,Approximate John’s decompositions, Operator Theory Advances and Applications77 (1995), 245–249.
[S1] S. Szarek,The finite dimensional basis problem, with an appendix on nets on Grassman manifold, Acta Mathematica151 (1983), 153–179.
[S2] S. Szazek, Spaces with large distance to ℓ n∞ and random matrices, American Journal of Mathematics112 (1990), 899–942.
[S-T] S. Szarek and M. Talagrand,An isomorphic version of the Sauer-Shelah lemma and the Banach-Mazur distance to the cube, Lecture Notes in Mathematics1376, Springer-Verlag, Berlin, 1989, pp. 105–112.
[T] M. Talagrand,Construction of majorizing measures, Bernoulli processes and cotype, Geometric and Functional Analysis4 (1994), 660–717.
[T-J] N. Tomczak-Jaegermann,Banach-Mazur Distances and Finite Dimensional Operator Ideals, Pitman Monographs,38, Longman, 1989.
Author information
Authors and Affiliations
Corresponding author
Additional information
Research supported in part by a grant of the US-Israel BSF.
Rights and permissions
About this article
Cite this article
Rudelson, M. Contact points of convex bodies. Isr. J. Math. 101, 93–124 (1997). https://doi.org/10.1007/BF02760924
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02760924