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

 

Coverings by convex bodies and inscribed balls


Author: Vladimir Kadets
Journal: Proc. Amer. Math. Soc. 133 (2005), 1491-1495
MSC (2000): Primary 52A37; Secondary 46C05
Published electronically: November 1, 2004
MathSciNet review: 2111950
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Abstract: Let $H$ be a Hilbert space. For a closed convex body $A$ denote by $r(A)$ the supremum of the radiuses of balls contained in $A$. We prove that $\sum_{n=1}^\infty r(A_n) \ge r(A)$ for every covering of a convex closed body $A \subset H$ by a sequence of convex closed bodies $A_n$, $n \in \mathbb{N} $. It looks like this fact is new even for triangles in a 2-dimensional space.


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Additional Information

Vladimir Kadets
Affiliation: Department of Mechanics and Mathematics, Kharkov National University, pl. Svobody 4, 61077 Kharkov, Ukraine
Address at time of publication: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
Email: vova1kadets@yahoo.com

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07650-6
PII: S 0002-9939(04)07650-6
Keywords: Hilbert space, convex sets, inscribed ball
Received by editor(s): November 6, 2003
Received by editor(s) in revised form: January 7, 2004
Published electronically: November 1, 2004
Additional Notes: The author expresses thanks to the Department of Mathematics, University of Missouri-Columbia, and especially to Professor Nigel Kalton for hospitality and a fruitful working atmosphere
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.