Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Convex bodies with few faces

Authors: Keith Ball and Alain Pajor
Journal: Proc. Amer. Math. Soc. 110 (1990), 225-231
MSC: Primary 52A40; Secondary 11H46, 52A20
MathSciNet review: 1019270
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It is proved that it $ {u_1}, \ldots ,{u_n}$ are vectors in $ {{\mathbf{R}}^k},k \leq n,1 \leq p < \infty $ and

$\displaystyle r = {\left( {\frac{1}{k}{{\sum\limits_1^n {\left\vert {{u_i}} \right\vert} }^p}} \right)^{1/p}}$

then the volume of the symmetric convex body whose boundary functionals are $ \pm {u_1}, \ldots , \pm {u_n}$, is bounded from below as

$\displaystyle {\left\vert {\left\{ {x \in {{\mathbf{R}}^k}:\left\vert {\left\la... ... \leq 1{\text{ for every }}i} \right\}} \right\vert^{1/k}} \geq 1/\sqrt \rho r.$

An application to number theory is stated.

References [Enhancements On Off] (What's this?)

  • [BF] I. Bárány and Z. Füredi, Computing the volume is difficult, Discrete Comput. Geom. 2 (1987), 319-326. MR 911186 (89a:68203)
  • [BLM] J. Bourgain, J. Lindenstrauss and V. D. Milman, Approximation of zonoids by zonotopes, Acta Math. 162 (1989), 73-141. MR 981200 (90g:46020)
  • [BP] K. M. Ball and A. Pajor, On the entropy of convex bodies with "few" extreme points, in preparation.
  • [BV] E. Bombieri and J. Vaaler, On Stegel's lemma, Invent. Math. 73 (1983), 11-32. MR 707346 (85g:11049a)
  • [CP] B. Carl and A. Pajor, Gelfand numbers of operators with values in a Hilbert space, Invent. Math. 94 (1988), 479-504. MR 969241 (90d:46023)
  • [FJ] T. Fiegiel and W. B. Johnson, Large subspaces of $ l_\infty ^n$ and estimates of the Gordon-Lewis constants, Israel J. Math. 37 (1980), 92-112. MR 599305 (81m:46031)
  • [G] E. D. Gluskin, Extremal properties of rectangular parallelipipeds and their applications to the geometry of Banach spaces, Mat. Sb. (N. S.) 136 (1988), 85-95.
  • [MeP] M. Meyer and A. Pajor, Sections of the unit ball of $ l_p^n$, J. Funct. Anal. 80 (1988), 109-123. MR 960226 (89h:52010)
  • [MiP] V. D. Milman and A. Pajor, Isotropie position and inertia ellipsoids and zonoids of the unit ball of a normed $ n$-dimensional space, Israel seminar on G.A.F.A. (1987-88), Springer-Verlag, Lectures Notes in Math., vol. 1376, 1989. MR 1008717 (90g:52003)
  • [PT] A. Pajor and N. Tomczak-Jaegermann, Subspaces of small codimension of finite-dimensional Banach spaces, Proc. Amer. Math. Soc. 97 (1986), 637-642. MR 845980 (87i:46040)
  • [S] C. Schütt, Entropy numbers of diagonal operators between symmetric Banach spaces, J. Approx. Theory 40 (1984), 121-128. MR 732693 (85e:47029)
  • [V] J. D. Vaaler, A geometric inequality with applications to linear forms, Pacific J. Math. 83 (1979), 543-553. MR 557952 (81d:52007)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 52A40, 11H46, 52A20

Retrieve articles in all journals with MSC: 52A40, 11H46, 52A20

Additional Information

Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society