Abstract. A hyperplane search tree is a binary tree used to store a set S of n d-dimensional data points. In a random hyperplane search tree for S, the root represents a hyperplane defined by d data points drawn uniformly at random from S. The remaining data points are split by the hyperplane, and the definition is used recursively on each subset. We assume that the data are points in general p...

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...

The addition-deletion theorems for hyperplane arrangements, which were originally shown in [T1], provide useful ways to construct examples of free arrangements. In this article, we prove addition-deletion theorems for multiarrangements. A key to the generalization is the definition of a new multiplicity, called the e-multiplicity, of a restricted multiarrangement. We compute the e-multiplicitie...

Given a set of n hyperplanes h1, . . . , hn ∈ R d the hyperplane depth of a point P ∈ R is the minimum number of hyperplanes that a ray from P can meet. The hyperplane depth of the arrangement is the maximal depth of points P not in any hi. We give an optimal O(n logn) deterministic algorithm to compute the hyperplane depth of an arrangement in dimension d = 2.