Median spheres: Theory, algorithms, applications (Q2267770)

The author deals with the problem of fitting circles and spheres to data which occur in archeological meters. Minimizing sums of first or second powers or minimizing the maximum of distances to the data leads to median, mean or midrange circles and spheres. In this framework, the author is focused on the goal to establish some relations between the geometry and the combinatorics of the median circles and spheres. 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 a finite set of data points. A weighted pseudo halving hypersphere is any hypersphere for which the absolute value of the difference between the sum of the weights of the data points inside and the same sum outside is less than the similar sum on the hypersphere. Combinatorically, a hypersphere is blocked by the data 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 the absolute value of the difference between \(k\) and \(l\) is less or equal to 1. The author proves that for each finite data set, there exists at least one weighted median generalized hypersphere. Also, each weighted median hypersphere is a weighted pseudo halving hypersphere and passes through at least two distinct data points. In the plane, the author shows 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 it remains in this way under sufficiently small perturbations of the data. Moreover, for every set of four points, at least one unweighted median circle is blocked. Finally, the author gives some algorithms based on theoretical results for median circles and discuss some applications related to some archeological problems and operations research problems.