A four--dimensional Neumann ovaloid

TL;DR

This paper constructs a four-dimensional Neumann ovaloid, explicitly solving an inverse gravitational potential problem for a domain of revolution that generalizes the planar Neumann oval.

Contribution

It provides the first explicit construction of a four-dimensional Neumann ovaloid solving the inverse potential problem.

Findings

Explicit four-dimensional Neumann ovaloid constructed.

Demonstrates existence and uniqueness of the domain.

Extends the concept from planar to four dimensions.

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.