Some Problems Involving Circular and Spherical Targets

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Abstract
This article is concerned with some problems that occur in certain tactical considerations: how should one place k circles [spheres] in the plane [3-space] so that their union has the greatest standard normal probability measure, that is, so as to maximize the probability that a random normal point will fall in one or more of the circles [spheres]. For k > 3 the problem seems hopeless, (except for certain special situations); the case for k = 3 is still unresolved and is being worked on by a number of investigators, and the case for k = 2 is solved completely in this paper. The results for k = 2 have some practical value when applied to actual problems arising in tactical considerations, and some theoretical value, as a method of attacking the problem for k ≧ 3.

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