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

   

 

Volume bounds for shadow covering


Authors: Christina Chen, Tanya Khovanova and Daniel A. Klain
Journal: Trans. Amer. Math. Soc. 366 (2014), 1161-1177
MSC (2010): Primary 52A20
Published electronically: August 20, 2013
MathSciNet review: 3145726
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Abstract: For $ n \geq 2$ a construction is given for a large family of compact convex sets $ K$ and $ L$ in $ \mathbb{R}^n$ such that the orthogonal projection $ L_u$ onto the subspace $ u^\perp $ contains a translate of the corresponding projection $ K_u$ for every direction $ u$, while the volumes of $ K$ and $ L$ satisfy $ V_n(K) > V_n(L).$

It is subsequently shown that if the orthogonal projection $ L_u$ onto the subspace $ u^\perp $ contains a translate of $ K_u$ for every direction $ u$, then the set $ \frac {n}{n-1} L$ contains a translate of $ K$. It follows that

$\displaystyle V_n(K) \leq \left (\frac {n}{n-1}\right )^n V_n(L).$

In particular, we derive a universal constant bound

$\displaystyle V_n(K) \leq 2.942 \, V_n(L),$

independent of the dimension $ n$ of the ambient space. Related results are obtained for projections onto subspaces of some fixed intermediate co-dimension. Open questions and conjectures are also posed.

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Additional Information

Christina Chen
Affiliation: Newton North High School, Newton, Massachusetts 02460

Tanya Khovanova
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139

Daniel A. Klain
Affiliation: Department of Mathematical Sciences, University of Massachusetts Lowell, Lowell, Massachusetts 01854
Email: Daniel\textunderscore{}Klain@uml.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05855-2
Received by editor(s): October 22, 2011
Published electronically: August 20, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.