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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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: 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,

$\displaystyle \vert P\tilde C\vert \geq \,\vert\tilde C{\vert^{(n - 1)\,/\,n}}.$

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 $ \vert K{\vert^{(n - 1)\,/\,n}}$.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1991-1035998-3
PII: S 0002-9947(1991)1035998-3
Article copyright: © Copyright 1991 American Mathematical Society