Abstract.
We prove that a bounded open set U in \({\mathbb{R}}^n\) 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 \({\mathbb{R}}^n\). 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.
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Received: January 2006, Revision: May 2006, Accepted: June 2006
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Guth, L. The Width-Volume Inequality. GAFA Geom. funct. anal. 17, 1139–1179 (2007). https://doi.org/10.1007/s00039-007-0628-5
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DOI: https://doi.org/10.1007/s00039-007-0628-5