Shadows of convex bodies
Author:
Keith Ball
Journal:
Trans. Amer. Math. Soc. 327 (1991), 891901
MSC:
Primary 52A40; Secondary 52A20
MathSciNet review:
1035998
Fulltext PDF Free Access
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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, It is also shown that there is a pathological body, , all of whose orthogonal projections have volume about times as large as .
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, Volumes ratios and a reverse isoperimetric inequality, In preparation.
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Bourgain and J.
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(89g:46024), http://dx.doi.org/10.1007/BFb0081746
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Herm
Jan Brascamp and Elliott
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and its generalization to more than three functions, Advances in Math.
20 (1976), no. 2, 151–173. MR 0412366
(54 #492)
 [BP]
H.
Busemann and C.
M. Petty, Problems on convex bodies, Math. Scand.
4 (1956), 88–94. MR 0084791
(18,922b)
 [FLM]
T.
Figiel, J.
Lindenstrauss, and V.
D. Milman, The dimension of almost spherical sections of convex
bodies, Acta Math. 139 (1977), no. 12,
53–94. MR
0445274 (56 #3618)
 [J]
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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
(52 #11737)
 [LW]
L.
H. Loomis and H.
Whitney, An inequality related to the
isoperimetric inequality, Bull. Amer. Math.
Soc 55 (1949),
961–962. MR 0031538
(11,166d), http://dx.doi.org/10.1090/S000299041949093205
 [MS]
Vitali
D. Milman and Gideon
Schechtman, Asymptotic theory of finitedimensional normed
spaces, Lecture Notes in Mathematics, vol. 1200, SpringerVerlag,
Berlin, 1986. With an appendix by M. Gromov. MR 856576
(87m:46038)
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M. Petty, Projection bodies, Proc. Colloquium on Convexity
(Copenhagen, 1965) Kobenhavns Univ. Mat. Inst., Copenhagen, 1967,
pp. 234–241. MR 0216369
(35 #7203)
 [R]
Shlomo
Reisner, Zonoids with minimal volumeproduct, Math. Z.
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Rolf
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Projektionen konvexer Körper, Math. Z. 101
(1967), 71–82 (German). MR 0218976
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Carsten
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Jeffrey
D. Vaaler, A geometric inequality with applications to linear
forms, Pacific J. Math. 83 (1979), no. 2,
543–553. MR
557952 (81d:52007)
 [B]
 K. M. Ball, Volumes of sections of cubes and related problems, Israel Seminar (G.A.F.A.) 1988, Lecture Notes in Math., vol. 1376, SpringerVerlag, Berlin and New York, 1989, pp. 251260. MR 1008726 (90i:52019)
 [B]
 , Volumes ratios and a reverse isoperimetric inequality, In preparation.
 [BL]
 J. Bourgain and J. Lindenstrauss, Projection bodies, Israel Seminar (G.A.F.A) 198687, Lecture Notes in Math., vol. 1317, SpringerVerlag, Berlin and New York, 1988, pp. 250269. MR 950986 (89g:46024)
 [BrL]
 Herm Jan Brascamp and Elliott H. Lieb, Best constants in Young's inequality, its converse and its generalization to more than three functions, Adv. in Math. 20 (1976), 151173. MR 0412366 (54:492)
 [BP]
 H. Busemann and C. M. Petty, Problems on convex bodies, Math. Scand. 4 (1956), 8894. MR 0084791 (18:922b)
 [FLM]
 T. Figiel, J. Lindenstrauss and V. D. Milman, The dimension of almost spherical sections of convex bodies, Acta Math. 139 (1977), 5394. MR 0445274 (56:3618)
 [J]
 F. John, Extremum problems with inequalities as subsidiary conditions, Courant Anniversary Volume, Interscience, New York, 1948, pp. 187204. MR 0030135 (10:719b)
 [LR]
 D. G. Larman and C. A. Rogers, The existence of a centrally symmetric convex body with central sections that are unexpectedly small, Mathematika 22 (1976), 164175. MR 0390914 (52:11737)
 [LW]
 L. H. Loomis and H. Whitney, An inequality related to the isoperimetric inequality, Bull. Amer. Math. Soc. 55 (1949), 961962. MR 0031538 (11:166d)
 [MS]
 V. D. Milman and G. Schechtman, Asymptotic theory of finite dimensional normed spaces, Lecture Notes in Math., vol. 1200, SpringerVerlag, Berlin and New York, 1986, pp. 64104. MR 856576 (87m:46038)
 [P]
 C. M. Petty, Projection bodies, Proc. Colloq. on Convexity, Copenhagen, 1967, pp. 234241. MR 0216369 (35:7203)
 [R]
 S. Reisner, Zonoids with minimal volume product, Math. Z. 192 (1986), 339346. MR 845207 (87g:52022)
 [S]
 R. Schneider, Zu einem Problem von Shephard über die Projektionen konvexer Körper, Math. Z. 101 (1967), 7182. MR 0218976 (36:2059)
 [Schü]
 C. Schütt, The isoperimetric quotient and some classical Banach spaces, (to appear). MR 1021360 (90j:46026)
 [V]
 J. D. Vaaler, A geometric inequality with applications to linear forms, Pacific J. Math. 83 (1979), 543553. MR 557952 (81d:52007)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199110359983
PII:
S 00029947(1991)10359983
Article copyright:
© Copyright 1991
American Mathematical Society
