The p-sphere and the geometric substratum of power-law probability distributions

Links between power law probability distributions and marginal distributions of uniform laws on p-spheres in Rn show that a mathematical derivation of the Boltzmann-Gibbs distribution necessarily passes through power law ones. Results are also given that link parameters p and n to the value of the non-extensivity parameter q that characterizes these power laws in the context of non-extensive statistics.