  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.
