标题： 一个四维Neumann椭球体 显示英文标题

标题： A four--dimensional Neumann ovaloid

摘要： 一个质量均匀分布的物体的形状是什么，它产生的引力势与两个相等的点质量产生的引力势相等？ 如果每个点质量的重量相对于两点之间的距离足够小，那么答案是两个半径相等的球体，每个球体分别以两个点为中心，否则它是一个围绕通过两点的轴旋转的某个区域。 这种区域的存在性和唯一性是已知的，但在平面上，该区域被称为Neumann椭圆，只有在平面上才有显式的参数化。 我们构建了一个四维的“Neumann椭圆体”，从而显式地解决了这个反向势问题。 显示英文摘要

摘要： 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.