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...

Finding nearest neighbor points in a large scale high dimensional data set is of wide interest in computer vision. One popular and efficient approach is to encode each data point as a binary code in Hamming space using separating hyperplanes. One condition which is often implicitly assumed is that the separating hyperplanes should be mutually orthogonal. With the aim of increasing the represent...

Given a Banach space we consider the $\sigma$-ideal of all its subsets which are covered by countably many hyperplanes and investigate standard cardinal characteristics as additivity, covering number, uniformity, cofinality. We determine their values for separable spaces, approximate them nonseparable spaces. The remaining questions reduce to deciding if following can be proved in ZFC (i.e. Zer...