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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

An inequality for sections and projections of a convex set


Author: Jonathan E. Spingarn
Journal: Proc. Amer. Math. Soc. 118 (1993), 1219-1224
MSC: Primary 52A39; Secondary 52A40
MathSciNet review: 1184087
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Abstract: Let $ K \subset {{\mathbf{R}}^d}$ be a convex body, $ \gamma $ its center of mass. For $ \Lambda \subset {{\mathbf{R}}^d}$ a subspace of dimension $ d - k$, we establish the inequality

$\displaystyle {\operatorname{Vol} _d}(K) \leqslant {\operatorname{Vol} _{d - k}}(K\vert\Lambda ){\operatorname{Vol} _k}((K - \gamma ) \cap {\Lambda ^ \bot })$

(where $ K\vert\Lambda $ denotes orthogonal projection of $ K$ onto $ \Lambda $). Equality holds only if each $ k$-dimensional section of $ K$ parallel to $ {\Lambda ^ \bot }$ is a translate of $ (K - \gamma ) \cap {\Lambda ^ \bot }$.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1993-1184087-3
PII: S 0002-9939(1993)1184087-3
Article copyright: © Copyright 1993 American Mathematical Society