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The width-volume inequality. (English) Zbl 1141.53039
Summary: We prove that a bounded open set \(U\) in \(\mathbb R^n\) has \(k\)-width less than \(C(n) \operatorname{Volume} (U)^{k/n}\). 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.

MSC:
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
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