Shadows of convex bodies

Ball, Keith

doi:10.1090/S0002-9947-1991-1035998-3

Shadows of convex bodies
HTML articles powered by AMS MathViewer

by Keith Ball PDF

Trans. Amer. Math. Soc. 327 (1991), 891-901 Request permission

Abstract:

It is proved that if $C$ is a convex body in ${\mathbb {R}^n}$ then $C$ has an affine image $\tilde C$ (of nonzero volume) so that if $P$ is any $1$-codimensional orthogonal projection, \[ |P\tilde C| \geq |\tilde C{|^{(n - 1) / n}}.\] It is also shown that there is a pathological body, $K$, all of whose orthogonal projections have volume about $\sqrt n$ times as large as $|K{|^{(n - 1) / n}}$.

References

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, DOI 10.1007/BFb0090058

Volumes ratios and a reverse isoperimetric inequality

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, DOI 10.1007/BFb0081746
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 412366, DOI 10.1016/0001-8708(76)90184-5
H. Busemann and C. M. Petty, Problems on convex bodies, Math. Scand. 4 (1956), 88–94. MR 84791, DOI 10.7146/math.scand.a-10457
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 445274, DOI 10.1007/BF02392234
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
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 390914, DOI 10.1112/S0025579300006033
L. H. Loomis and H. Whitney, An inequality related to the isoperimetric inequality, Bull. Amer. Math. Soc 55 (1949), 961–962. MR 0031538, DOI 10.1090/S0002-9904-1949-09320-5
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
C. M. Petty, Projection bodies, Proc. Colloquium on Convexity (Copenhagen, 1965) Kobenhavns Univ. Mat. Inst., Copenhagen, 1967, pp. 234–241. MR 0216369
Shlomo Reisner, Zonoids with minimal volume-product, Math. Z. 192 (1986), no. 3, 339–346. MR 845207, DOI 10.1007/BF01164009
Rolf Schneider, Zur einem Problem von Shephard über die Projektionen konvexer Körper, Math. Z. 101 (1967), 71–82 (German). MR 218976, DOI 10.1007/BF01135693
Carsten Schütt, The isoperimetric quotient and some classical Banach spaces, Israel J. Math. 67 (1989), no. 1, 43–61. MR 1021360, DOI 10.1007/BF02764898
Jeffrey D. Vaaler, A geometric inequality with applications to linear forms, Pacific J. Math. 83 (1979), no. 2, 543–553. MR 557952