The Width-Volume Inequality |
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Authors: | Larry Guth |
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Institution: | (1) Department of Mathematics, Stanford, Stanford, CA 94305, USA |
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Abstract: | We prove that a bounded open set U in has k-width less than C(n) Volume(U)
k/n
. Using this estimate, we give lower bounds for the k-dilation of degree 1 maps between certain domains in . In particular, we estimate the smallest (n – 1)-dilation of any degree 1 map between two n-dimensional rectangles. For any pair of rectangles, our estimate is accurate up to a dimensional constant C(n). We give examples in which the (n – 1)-dilation of the linear map is bigger than the optimal value by an arbitrarily large factor.
Received: January 2006, Revision: May 2006, Accepted: June 2006 |
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Keywords: | Sweepout k-dilation k-width |
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