The gaussian centre and the projection centre of a set of points in r3

! #"%$& ' () Finding a centre point is a fundamental problem of geometry. The Euclidean centre, or centre of the smallest enclosing sphere, provides a natural definition for the centre of a set of points. As shown in [BBKS00] and [DK04], the Euclidean centre of a set of points * + -, is unstable; small perturbations at only a few points of can result in an arbitrarily large relative change in the position of the Euclidean centre. To define a centre . more stable than the Euclidean centre requires, at least for some sets of points, that . differ from the Euclidean centre. Presumably, remaining central to is desirable. These two factors are in opposition; high stability implies high eccentricty and vice-versa. In [DK04], the Gaussian centre of a set of points in the plane is introduced toward the objective of identifying a good centre that balances high stability with low eccentricty. The projection centre of a set of points in the plane is also defined and shown to be equivalent to the Gaussian centre. The Gaussian centre’s benefits extend beyond its definition as the centre of a set of static points. Recently, several questions of facility location have been posed within the setting of mobile facility location (e.g. [AGG02, AH01, BBKS00, Her03]). Given a set of mobile points, the fitness of a mobile facility is determined both by its eccentricity and also by the maximum velocity and continuity of its motion. As shown in [DK04], the stability of a centre is inversely related to the maximum velocity of a mobile facility, providing further motivation for the need of stability in a centre point. The question of whether the Gaussian centre generalizes to three dimensions remained open. In this paper we define the Gaussian centre and the projection centre of a set of points / 0 . We show the equivalence of the two centres for any non-empty finite set .