The Shi arrangement Sn is the arrangement of affine hyperplanes in R n of the form xi−xj = 0 or 1, for 1 ≤ i < j ≤ n. It dissects R n into (n+1) regions, as was first proved by Shi. We give a simple bijective proof of this result. Our bijection generalizes easily to any subarrangement of Sn containing the hyperplanes xi − xj = 0 and to the extended Shi arrangements.

Let ∆ be a dual polar space of rank n ≥ 4, H be a hyperplane of ∆ and Γ := ∆\H be the complement of H in ∆. We shall prove that, if all lines of ∆ have more than 3 points, then Γ is simply connected. Then we show how this theorem can be exploited to prove that certain families of hyperplanes of dual polar spaces, or all hyperplanes of certain dual polar spaces, arise from embeddings.

Basic properties of finite subsets of the integer lattice Z are investigated from the point of view of geometric tomography. Results obtained concern the Minkowski addition of convex lattice sets and polyominoes, discrete X-rays and the discrete and continuous covariogram, the determination of symmetric convex lattice sets from the cardinality of their projections on hyperplanes, and a discrete...

It is NP complete to recognize whether two sets of points in general space can be separated by two hyperplanes It is NP complete to recognize whether two sets of points in the plane can be separated with k lines For every xed k in any xed dimension it takes polynomial time to recognize whether two sets of points can be separated with k hyperplanes

From the above definition, we have the following geometrical interpretation of the dual lattice. For any vector x, the set of all points whose inner product with x is integer forms a set of hyperplanes perpendicular to x and separated by distance 1/‖x‖. Hence, any vector x in a lattice Λ imposes the constraint that all points in Λ∗ lie in one of the hyperplanes defined by x. See the next figure...

The Nevanlinna defect relation and other results on the Nevanlinna theory assert that each meromorphic mapping f of C into P(C) has few deficient hyperplanes in P(C) . However, it seems to me that meromorphic mappings with a deficient hyperplane must be very few. In this paper, we show that for an arbitrary given transcendental meromorphic mapping f (which may be linearly degenerate), we can el...

The paper is devoted to computation of the cohomology of the complex of logarithmic differential forms with coefficients in rational functions whose poles are located on the union of several hyperplanes of a linear space over a field of characteristic zero. The main result asserts that for a vast class of hyperplane arrangements, including all free and generic arrangements, the cohomology algeb...

We study an NP-complete geometric covering problem called d-Dimensional Rectangle Stabbing, where, given a set of axis-parallel d-dimensional hyperrectangles, a set of axis-parallel (d − 1)-dimensional hyperplanes and a positive integer k, the question is whether one can select at most k of the hyperplanes such that every hyperrectangle is intersected by at least one of these hyperplanes. This ...

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...

Data envelopment analysis (DEA) is a non-parametric method for evaluating the relative technical efficiency for each member of a set of peer decision making units (DMUs) with multiple inputs and multiple outputs. The original DEA models use positive input and output variables that are measured on a ratio scale, but these models do not apply to the variables in which interval scale data can appe...