Approximation of Distributions by Sets

The well-known ’k-means’ clustering can be regarded as an approximation of a given distribution (which can be a sample) by a set of optimally chosen k points. However, in many cases approximative sets of different types are of interest. For example, approximation of a distribution by circles is important in allocating communication stations, the circles being interpreted as working areas of the stations. The paper covers two related topics. First we propose a heuristic algorithm to find k circles of a given radius r that fit with the planar data set. Then we analyse the problem of consistency: does a sequence of sample-based sets of optimal circles converge to the class of optimal circles for the population? The positive answer is given for arbitrary finite-dimensional normed linear spaces.