The Width-volume Inequality

We prove that a bounded open set U in 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 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. This paper proves some estimates having to do with the areas of k-dimensional surfaces in Euclidean space. We deal with two problems. First, suppose that U is a bounded open set in R n. We consider the problem of sweeping out U with k-dimensional surfaces, trying to arrange that the volumes of all the surfaces are as small as possible. Depending on the geometry of U , we give upper and lower bounds for the possible volumes of the surfaces. In particular, we construct a family of k-dimensional surfaces sweeping out U so that each surface has volume bounded by C(n)Volume(U) k/n. The next question concerns mappings from one open set to another-for example from the unit cube to a long thin cylinder. After we fix a domain and a range, we consider the problem of finding a degree 1 mapping which stretches the volumes of k-dimensional surfaces as little as possible. For certain pairs of (n-dimensional) rectangles, we show that the linear mapping stretches the k-dimensional surfaces far more than necessary. We give upper and lower bounds for the minimal amount of stretching by any degree 1 map. When k = n − 1, these upper and lower bounds match up to a constant factor. The definition of k-width As a first approximation to the definition of width, we define a linear version of k-dimensional width. Let U be a bounded open set in R n. For each (n-k)-plane P through the origin, let F (P) denote the family of all k-planes perpendicular to P. Define the width of F (P) to be the maximum volume of the intersection of U with any of the k-planes in F (P). Then define the linear k-width of U to be the minimum width of F (P) as P varies among all the (n-k)-planes through the origin. The width considered in …