Uniform Distributions on Spheres in Finite DimensionalLαand Their Generalizations

We characterize uniform distributions on spheres in n-dimensional spacesLαby certain Cauchy-like (n−1)-dimensional distributions of the quotients and derive some properties of mixtures of uniform distributions on such spheres, i.e.,α-spherical distributions. We feel that a simple Cauchy-like distribution is much simpler to deal with than the usual description of a uniform distribution on the sphere.