## Convex bodies with few faces

HTML articles powered by AMS MathViewer

- by Keith Ball and Alain Pajor
- Proc. Amer. Math. Soc.
**110**(1990), 225-231 - DOI: https://doi.org/10.1090/S0002-9939-1990-1019270-8
- PDF | Request permission

## 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 \[ r = {\left ( {\frac {1}{k}{{\sum \limits _1^n {\left | {{u_i}} \right |} }^p}} \right )^{1/p}}\] then the volume of the symmetric convex body whose boundary functionals are $\pm {u_1}, \ldots , \pm {u_n}$, is bounded from below as \[ {\left | {\left \{ {x \in {{\mathbf {R}}^k}:\left | {\left \langle {x,{u_i}} \right \rangle } \right | \leq 1{\text { for every }}i} \right \}} \right |^{1/k}} \geq 1/\sqrt \rho r.\] An application to number theory is stated.## References

- Imre Bárány and Zoltán Füredi,
*Computing the volume is difficult*, Discrete Comput. Geom.**2**(1987), no. 4, 319–326. MR**911186**, DOI 10.1007/BF02187886 - J. Bourgain, J. Lindenstrauss, and V. Milman,
*Approximation of zonoids by zonotopes*, Acta Math.**162**(1989), no. 1-2, 73–141. MR**981200**, DOI 10.1007/BF02392835
K. M. Ball and A. Pajor, - E. Bombieri and J. Vaaler,
*On Siegel’s lemma*, Invent. Math.**73**(1983), no. 1, 11–32. MR**707346**, DOI 10.1007/BF01393823 - Bernd Carl and Alain Pajor,
*Gel′fand numbers of operators with values in a Hilbert space*, Invent. Math.**94**(1988), no. 3, 479–504. MR**969241**, DOI 10.1007/BF01394273 - T. Figiel and W. B. Johnson,
*Large subspaces of $l^{n}_{\infty }$ and estimates of the Gordon-Lewis constant*, Israel J. Math.**37**(1980), no. 1-2, 92–112. MR**599305**, DOI 10.1007/BF02762871
E. D. Gluskin, - Mathieu Meyer and Alain Pajor,
*Sections of the unit ball of $L^n_p$*, J. Funct. Anal.**80**(1988), no. 1, 109–123. MR**960226**, DOI 10.1016/0022-1236(88)90068-7 - V. D. Milman and A. Pajor,
*Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed $n$-dimensional space*, Geometric aspects of functional analysis (1987–88), Lecture Notes in Math., vol. 1376, Springer, Berlin, 1989, pp. 64–104. MR**1008717**, DOI 10.1007/BFb0090049 - Alain Pajor and Nicole Tomczak-Jaegermann,
*Subspaces of small codimension of finite-dimensional Banach spaces*, Proc. Amer. Math. Soc.**97**(1986), no. 4, 637–642. MR**845980**, DOI 10.1090/S0002-9939-1986-0845980-8 - Carsten Schütt,
*Entropy numbers of diagonal operators between symmetric Banach spaces*, J. Approx. Theory**40**(1984), no. 2, 121–128. MR**732693**, DOI 10.1016/0021-9045(84)90021-2 - Jeffrey D. Vaaler,
*A geometric inequality with applications to linear forms*, Pacific J. Math.**83**(1979), no. 2, 543–553. MR**557952**

*On the entropy of convex bodies with "few" extreme points*, in preparation.

*Extremal properties of rectangular parallelipipeds and their applications to the geometry of Banach spaces*, Mat. Sb. (N. S.)

**136**(1988), 85-95.

## Bibliographic Information

- © Copyright 1990 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**110**(1990), 225-231 - MSC: Primary 52A40; Secondary 11H46, 52A20
- DOI: https://doi.org/10.1090/S0002-9939-1990-1019270-8
- MathSciNet review: 1019270