This paper gives several new demonstrations of the usefulness of random sampling techniques in computational geometry. One new algorithm creates a search structure for arrangements of hyperplanes by sampling the hyperplanes and using information from the resulting arrangement to divide and conquer. This algorithm requires O(s) expected preprocessing time to build a search structure for an arran...

We study multiplicative Diophantine approximation property of vectors and compute Diophantine exponents of hyperplanes via dynamics.

In this paper we deal with the location of hyperplanes in n{ dimensional normed spaces. If d is a distance measure, our objective is to nd a hyperplane H which minimizes points and d(x m ; H) = min z2H d(x m ; z) is the distance from x m to the hyperplane H. In robust statistics and operations research such an optimal hyperplane is called a median hyperplane. We show that for all distance measu...

We fix three natural numbers k, n,N , such that n + k + 1 = N , and introduce the notion of two dual arrangements of hyperplanes. One of the arrangements is an arrangement of N hyperplanes in a k-dimensional affine space, the other is an arrangement of N hyperplanes in an n-dimensional affine space. We assign weights α1, . . . , αN to the hyperplanes of the arrangements and for each of the arra...

Is Hartwick’s rule a necessary condition for maximin in Solow’s [1974] model? Until recently this has been an open question; this is surprising given the prominence of the model. Cairns and Yang [2000] as well as Withagen, Asheim and Buchholz (this issue) claim that the answer is in the affirmative and claim to provide a formal proof. The latter team argues that the proof by the former is not c...

Let K be a field of characteristic zero, α algebraic of degree n over K. Given a proper parametrization ψ of a rational curve C, we present a new algorithm to compute the hypercircle associated to the parametrization ψ. As a consequence, we can decide if C is defined over K and, if not, to compute the minimum field of definition of C containing K. The algorithm exploits the conjugate curves of ...

We examine the existence of tangent hyperplanes to subriemannian balls. Strictly abnormal shortest paths are allowed.

Let S(N) ≡ PG(1, 2)×PG(1, 2)×· · ·×PG(1, 2) be a Segre variety that is N -fold direct product of projective lines of size three. Given two geometric hyperplanes H ′ and H ′′ of S(N), let us call the triple {H , H , H ∆H } the Veldkamp line of S(N). We shall demonstrate, for the sequence 2 ≤ N ≤ 4, that the properties of geometric hyperplanes of S(N) are fully encoded in the properties of Veldka...

We analyze the properties of an interior point cutting plane algorithm that is based on a semi-infinite linear formulation of the dual semidefinite program. The cutting plane algorithm approximately solves a linear relaxation of the dual semidefinite program in every iteration and relies on a separation oracle that returns linear cutting planes. We show that the complexity of a variant of the i...