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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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An inequality for sections and projections of a convex set
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by Jonathan E. Spingarn PDF
Proc. Amer. Math. Soc. 118 (1993), 1219-1224 Request permission

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 \[ {\operatorname {Vol} _d}(K) \leqslant {\operatorname {Vol} _{d - k}}(K|\Lambda ){\operatorname {Vol} _k}((K - \gamma ) \cap {\Lambda ^ \bot })\] (where $K|\Lambda$ 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
  • © Copyright 1993 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 118 (1993), 1219-1224
  • MSC: Primary 52A39; Secondary 52A40
  • DOI: https://doi.org/10.1090/S0002-9939-1993-1184087-3
  • MathSciNet review: 1184087