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 $\mathbb{R}^{n}$ 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.