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 ∈ 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)) where w(K) is the Gaussian mean width of K. Using the map that sends x ∈ K to the sign vector with respect to the hyperplanes, we conclude that every bounded subset K of R embeds into the Hamming cube {−1, 1} with a small distortion in the Gromov-Haussdorff 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.