Largest empty sphere

Abstract: A key problem in computational geometry is the identification of subsets of a point set having particular properties. We study this problem for the properties of convexity and emptiness. We show that finding empty triangles is related to the problem of determining pairs of vertices that see each other in a star-shaped polygon. A linear time algorithm for this problem which is of independent interest yields an optimal algorithm for finding all empty triangles. This result is then extended to an algorithm for finding empty convex r-gons (r > 3) and for determining a largest empty convex subset. Finally, extensions to higher dimensions are mentioned.

Abstract: This paper outlines the following generalization of the classical maximal-empty-rectangle (MER) problem: given n arbitrarily-oriented non-intersecting line segments of finite length on a rectangular floor, locate an empty isothetic rectangle of maximum area. Thus, the earlier restriction on isotheticity of the obstacles is relaxed. Based on the wellknown technique of matrix searching, a novel algorithm of time complexity O(nlog2n) and space complexity O(n), is proposed. Next, the technique is extended to handle the following two related open problems: locating the largest isothetic MER (i) inside an arbitrary simple polygon and (ii) amidst a set of arbitrary polygonal obstacles.

Abstract: A generalized hypersphere is either a hyperplane or a hypersphere, which consists of all points equidistant from a center. Geometrically, a weighted median hypersphere minimizes a weighted average of the distances from it to finitely many data points. As proved here, for each finite data set there exists at least one weighted median generalized hypersphere. Moreover, denote the sums of the weights of the data points inside by W −, outside by W +, and on the hypersphere by W 0. The present results show that each weighted median hypersphere is a weighted pseudo-halving hypersphere, in the sense that |W − − W +| < W 0, and passes through at least two distinct data points. Combinatorically, a hypersphere is blocked if and only if it passes through data points in general position, in the sense that no other hypersphere passes through the same data points. A hypersphere is a halving hypersphere if and only if it is blocked, contains exactly k data points inside, confines exactly l data points outside, and |k − l| ≤ 1. In the plane, the present results also show that if a median circle is not a halving circle, then moving its center along a median between two data points on it until it passes through the next data point yields a halving circle. Relative to the center, if the direction cosines of the external and internal data points have the same mean and variance, then the median circle must be blocked, and stays so under sufficiently small perturbations of the data. Moreover, for every set of four points, at least one unweighted median circle is blocked. These results lend credence to a variant of a method used by archaeologists, and explain some findings from operations research.