Circumspheres of sets of n + 1 random points in the d-dimensional Euclidean unit ball (1 ≤ n ≤ d)

Abstract

In the d -dimensional Euclidean space, any set of 1 n  independent random points, uniformly distributed in the interior of a unit ball of center O , determines almost surely a circumsphere of center C and of radius         and 1 T    . The rv  has a simple geometrical meaning which appears in figure 1a. A non-degenerate triangle ' O CM   0, 0   , where M is any point of the circumsphere except B , yields '' O M O C CM     . If M coincides with B , then the triangle becomes a segment and ' OB   . Therefore, a circumsphere is contained in the unit ball if and only if r   . We notice, en passant, that the rv's  , C  ,  are also relevant for circumspheres which cut   1 1 d S . We shall derive, for any 1 d  and any 1 nd and for circumspheres of () n d C , the joint probability density function (pdf) of the length  and of the circumradius  as well as the associated marginal pdf's.For reasons discussed below, the case of two random points   the length 12 AA has been calculated during the last hundred years by a variety of methods which yield different formal expressions [6, 15-16, 18-20, 24-25, 30, 36, 40, 44, 46, 48-49, 52, 58, 60-61, 63].The pdf's of the distance between two points randomly distributed in the inside of spheres or of ellipsoids find numerous applications. For instance, they were shown to simplify the calculation of self-energies of spherically symmetric matter distributions interacting by means of radially symmetric two-body potentials [48][49]. These calculations were extended to ellipsoids as a first step towards convex bodies whose shapes deviate from spherical. García-Pelayo [24] derived the distance pdf in ellipsoids as an integral he applied to a study of the shape of the earth. Other physical applications of distance pdf's include in particular the use of double electron-electron resonance to study spherical aggregates with shell structure [34] and the field of wireless networks whose properties are strongly influenced by distances between nodes [45][46]58]. Finally, it is worth mentioning the connection between distance pdf's and pdf's of random chord length in convex bodies, which depends on the considered secant randomness (see for instance [13]). The chord length pdf's apply for instance in the fields of neutronics and of reactor physics [19, 26, 35, 37, 41-42, 52, 57, 63]. Extensions of the previous problem include for instance the determination of the mean distance between a reference point and its n th neighbour among a collection of N points uniformly distributed in a hypersphere or in a hypercube of unit volumes in a d -dimensional Euclidean space [8]. Applying the results of the latter authors, Kowalski [38] gave a geometrical interpretation to a generalization of a distribution used to represent pion distribution in hadronic production models. Circumcircles and circumspheres play an important role in computational geometry. Domains of all kinds are meshed with Delaunay triangulations and Voronoi tessellations are constructed from them. Direct ap...