Geometric and Functional Analysis

, Volume 17, Issue 4, pp 1139–1179 | Cite as

The Width-Volume Inequality

  • Larry GuthEmail author


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.

Keywords and phrases:

Sweepout k-dilation k-width 

AMS Mathematics Subject Classification:



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Copyright information

© Birkhaeuser 2007

Authors and Affiliations

  1. 1.Department of Mathematics, StanfordStanfordUSA

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