Abstract A simple geometric mechanism: ‘the locus of intersections perpendicular bisectors and normal lines’, often arises in many guises nonlinear sciences. In this paper, a new application mechanism is given. Namely, we show that gives answers to all four basic problems on envelopes created by hyperplane families (existence problem, representation equivalence problem definitions, uniqueness p...

L1 norm discrimination consists in finding the hyperplane that minimizes the sum of L1 norm distances between the hyperplane and the points that lie on the wrong side of the hyperplane. This problem is difficult for datasets containing more than 100,000 points. Since few points are needed to obtain the optimal hyperplane, we propose a point selection algorithm which iteratively adds the necessa...

A binary space partitioning (bsp) tree represents a recursive, hierarchical subdivision of d-dimensional space into convex subspaces. BSP trees provide a computational representation of space that provides both a search structure and a representation of geometry. Constructing a bsp tree consists of partitioning a subspace by means of a hyperplane, which intersects the interior of that subspace,...

Two notions of riffle shuffling on finite Coxeter groups are given: one using Solomon’s descent algebra and another using random walk on chambers of hyperplane arrangements. These coincide for types A,B,C, H3, and rank two groups. Both notions have the same, simple eigenvalues. The hyperplane definition is especially natural and satisfies a positivity property when W is crystallographic and the...

The anchored hyperplane location problem is to locate a hyperplane passing through some given points P ⊆ IR and minimizing either the sum of weighted distances (median problem), or the maximum weighted distance (center problem) to some other points Q ⊆ IR. This problem of computational geometry is analyzed by using nonlinear programming techniques. If the distances are measured by a norm, it wi...

R. Thomas (with a remark of B. Totaro) proved that the Hodge conjecture is essentially equivalent to the existence of a hyperplane section, called a generalized Thomas hyperplane section, such that the restriction to it of a given primitive Hodge class does not vanish. We study the relations between the vanishing cycles in the cohomology of a general fiber, and show that each relation between t...