By introducing redundant Klee-Minty examples, we have previously shown that the central path can be bent along the edges of the Klee-Minty cubes, thus having 2n − 2 sharp turns in dimension n. In those constructions the redundant hyperplanes were placed parallel with the facets active at the optimal solution. In this paper we present a simpler and more powerful construction, where the redundant...

Let A be a real arrangement of hyperplanes. Let B = B(q) be Varchenko’s quantum bilinear form of A, introduced [15], specialized so that all hyperplanes have weight q . B(q) is nonsingular for all complex q except certain roots of unity. Here, we examine the kernel of B at roots of unity in relation to the topology of the hyperplane singularity. We use Varchenko’s work [16] to relate B(q) to a ...

We prove a new, tight upper bound on the number of incidences between points and hyperplanes in Euclidean d-space. Given n points, of which k are colored red, there are Od(m 2/3 k 2/3 n +kn+m) incidences between the k red points and m hyperplanes spanned by all n points provided that m = Ω(n). For the monochromatic case k = n, this was proved by Agarwal and Aronov [1]. We use this incidence bou...

In this paper we present some theoretical results on two forms of multi-point crossover: n-point crossover and uniform crossover. This analysis extends the work from De long's thesis, which dealt with dis~ption of n-point crossover on 2nd order hyperplanes. We present various extensions to this theory, including 1) an analysis of the disruption of n-point crossover on kth order hyperplanes: 2) ...

This work is motivated by several basic problems and techniques that rely on space decomposition of arrangements of hyperplanes in high-dimensional spaces, most notably Meiser’s 1993 algorithm for point location in such arrangements. A standard approach to these problems is via random sampling, in which one draws a random sample of the hyperplanes, constructs a suitable decomposition of its arr...