A four-dimensional Neumann ovaloid

Abstract

Abstract.What is the shape of a uniformly massive object that generates a gravitational potential equivalent to that of two equal point-masses? If the weight of each point-mass is sufficiently small compared to the distance between the points then the answer is a pair of balls of equal radius, one centered at each of the two points, but otherwise it is a certain domain of revolution about the axis passing through the two points. The existence and uniqueness of such a domain is known, but an explicit parameterization is known only in the plane where the region is referred to as a Neumann oval. We construct a four-dimensional "Neumann ovaloid", solving explicitly this inverse potential problem.

Neutral inclusions, weakly neutral inclusions, and an over-determined problem for confocal ellipsoids

“…These two-dimensional Neumann ovals were discovered by C. Neumann [33]. The existence and uniqueness of the higher dimensional Neumann ovaloids are known (see [22] and references therein), but there is no known explicit expression except four-dimensional (and two-dimensional) ones, to the best of our knowledge. We refer to a recent paper [22] for an explicit parametrization of a four-dimensional Neumann ovaloid.…”