We consider packings of radius r collars about hyperplanes in H. For such packings, we prove that the Delaunay cells are truncated ultra-ideal simplices which tile H. If we place n+1 hyperplanes in H each at a distance of exactly 2r to the others, we could place radius r collars about these hyperplanes. The density of these collars within the corresponding Delaunay cell is an upper bound on den...

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...

We present a branch-and-bound algorithm for minimizing multiple convex quadratic objective functions over integer variables. Our method looks efficient points by fixing subsets of variables to values and using lower bounds in the form hyperplanes image space derived from continuous relaxations restricted functions. show that stops after finitely many fixings with detecting both full nondominate...

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...