The width-volume inequality

Details The width-volume inequality The width-volume inequality

2006-09-20

arxiv.org/abs/math/0609569v1

Mathematics

Differential Geometry

31 pages, 4 figures, to be published in Geometric and Functional Analysis

Scientific paper

We prove that a bounded open set U in Euclidean n-space 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 Euclidean space. 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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