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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
##### Keywords:
sweepout; $$k$$-dilation; $$k$$-width
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