Dimension reduction by random hyperplane tessellations

Given a subset K of the unit Euclidean sphere, we estimate the minimal number m = m(K) of hyperplanes that generate a uniform tessellation of K, in the sense that the fraction of the hyperplanes separating any pair x, y in K is nearly proportional to the Euclidean distance between x and y. Random hyperplanes prove to be almost ideal for this problem; they achieve the almost optimal bound m = O(w(K)^2) where w(K) is the Gaussian mean width of K. Using the map that sends x in K to the sign vector with respect to the hyperplanes, we conclude that every bounded subset K of R^n embeds into the Hamming cube {-1, 1}^m with a small distortion in the Gromov-Haussdorf metric. Since for many sets K one has m = m(K) << n, this yields a new discrete mechanism of dimension reduction for sets in Euclidean spaces.Comment: 17 pages, 3 figures, minor update

Quasi-isometries Between Tubular Groups

We give a method of constructing maps between tubular groups inductively according to a set of strategies. This map will be a quasi-isometry exactly when the set of strategies is consistent. Conversely, if there exists a quasi-isometry between tubular groups, then there is a consistent set of strategies for them. There is an algorithm that will in finite time either produce a consistent set of strategies or decide that such a set does not exist. Consequently, this algorithm decides whether or not the groups are quasi-isometric.Comment: 44 pages, 11 figures. PDFLaTeX. Improved exposition and added some auxiliary material to make the paper more self contained, per referee's comment