  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.
