We prove existence of equal area partitions of the unit sphere via optimal transport methods, accompanied by diameter bounds written in terms of Monge--Kantorovich distances. This can be used to obtain bounds on the expectation of the maximum diameter of partition sets, when points are uniformly sampled from the sphere. An application to the computation of sliced Monge--Kantorovich distances is also presented.

翻译：我们通过最优传输方法证明了单位球面上存在具有直径边界的等面积划分，且直径边界可用Monge-Kantorovich距离表示。该结果可用于推导球面上均匀采样点时划分集最大直径的期望边界。同时给出了该方法在切片Monge-Kantorovich距离计算中的应用。