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Proceedings of the American Mathematical Society

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

 

 

Slicing convex bodies--bounds for slice area in terms of the body's covariance


Author: Douglas Hensley
Journal: Proc. Amer. Math. Soc. 79 (1980), 619-625
MSC: Primary 52A40
DOI: https://doi.org/10.1090/S0002-9939-1980-0572315-5
MathSciNet review: 572315
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Abstract: Let Q be a zero-symmetric convex set in $ {{\mathbf{R}}^N}$ with volume 1 and covariance matrix $ V^2 \mathrm{Id}_{N \times N}$. Let P be a K-dimensional vector subspace of $ {{\mathbf{R}}^n},K < N$, and let $ J = N - K$. Then there exist constants $ {C_1}(J)$ and $ {C_2}(J)$ such that

$\displaystyle {V^{ - J}}{C_1}(J) \leqslant \mathrm{vol}_K(P \cap Q) \leqslant V^{-J}{C_2}(J).$

The lower bound has applications to Diophantine equations.

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DOI: https://doi.org/10.1090/S0002-9939-1980-0572315-5
Keywords: Log concave, convex body, covariance, slice area bounds
Article copyright: © Copyright 1980 American Mathematical Society