Shadows of convex bodies

Author:
Keith Ball

Journal:
Trans. Amer. Math. Soc. **327** (1991), 891-901

MSC:
Primary 52A40; Secondary 52A20

MathSciNet review:
1035998

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Abstract | References | Similar Articles | Additional Information

Abstract: It is proved that if is a convex body in then has an affine image (of nonzero volume) so that if is any -codimensional orthogonal projection,

**[B]**Keith Ball,*Volumes of sections of cubes and related problems*, Geometric aspects of functional analysis (1987–88), Lecture Notes in Math., vol. 1376, Springer, Berlin, 1989, pp. 251–260. MR**1008726**, 10.1007/BFb0090058**[B]**-,*Volumes ratios and a reverse isoperimetric inequality*, In preparation.**[B-L]**J. Bourgain and J. Lindenstrauss,*Projection bodies*, Geometric aspects of functional analysis (1986/87), Lecture Notes in Math., vol. 1317, Springer, Berlin, 1988, pp. 250–270. MR**950986**, 10.1007/BFb0081746**[Br-L]**Herm Jan Brascamp and Elliott H. Lieb,*Best constants in Young’s inequality, its converse, and its generalization to more than three functions*, Advances in Math.**20**(1976), no. 2, 151–173. MR**0412366****[B-P]**H. Busemann and C. M. Petty,*Problems on convex bodies*, Math. Scand.**4**(1956), 88–94. MR**0084791****[F-L-M]**T. Figiel, J. Lindenstrauss, and V. D. Milman,*The dimension of almost spherical sections of convex bodies*, Acta Math.**139**(1977), no. 1-2, 53–94. MR**0445274****[J]**Fritz John,*Extremum problems with inequalities as subsidiary conditions*, Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, Interscience Publishers, Inc., New York, N. Y., 1948, pp. 187–204. MR**0030135****[L-R]**D. G. Larman and C. A. Rogers,*The existence of a centrally symmetric convex body with central sections that are unexpectedly small*, Mathematika**22**(1975), no. 2, 164–175. MR**0390914****[L-W]**L. H. Loomis and H. Whitney,*An inequality related to the isoperimetric inequality*, Bull. Amer. Math. Soc**55**(1949), 961–962. MR**0031538**, 10.1090/S0002-9904-1949-09320-5**[M-S]**Vitali D. Milman and Gideon Schechtman,*Asymptotic theory of finite-dimensional normed spaces*, Lecture Notes in Mathematics, vol. 1200, Springer-Verlag, Berlin, 1986. With an appendix by M. Gromov. MR**856576****[P]**C. M. Petty,*Projection bodies*, Proc. Colloquium on Convexity (Copenhagen, 1965) Kobenhavns Univ. Mat. Inst., Copenhagen, 1967, pp. 234–241. MR**0216369****[R]**Shlomo Reisner,*Zonoids with minimal volume-product*, Math. Z.**192**(1986), no. 3, 339–346. MR**845207**, 10.1007/BF01164009**[S]**Rolf Schneider,*Zur einem Problem von Shephard über die Projektionen konvexer Körper*, Math. Z.**101**(1967), 71–82 (German). MR**0218976****[Schü]**Carsten Schütt,*The isoperimetric quotient and some classical Banach spaces*, Israel J. Math.**67**(1989), no. 1, 43–61. MR**1021360**, 10.1007/BF02764898**[V]**Jeffrey D. Vaaler,*A geometric inequality with applications to linear forms*, Pacific J. Math.**83**(1979), no. 2, 543–553. MR**557952**

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DOI:
http://dx.doi.org/10.1090/S0002-9947-1991-1035998-3

Article copyright:
© Copyright 1991
American Mathematical Society