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)²) 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 $mathbb{R}^{n}$ embeds into the Hamming cube {?1,1} ^m 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.