Let $\bX=(X_1, \hdots, X_d)$ be a $\mathbb R^d$-valued random vector with i.i.d.~components, and let $\Vert\bX\Vert_p= ( \sum_{j=1}^d|X_j|^p)^{1/p}$ be its $p$-norm, for $p>0$. The impact of letting $d$ go to infinity on $\Vert\bX\Vert_p$ has surprising consequences, which may dramatically affect high-dimensional data processing. This effect is usually referred to as the {\it distance concentration phenomenon} in the computational learning literature. Despite a growing interest in this important question, previous work has essentially characterized the problem in terms of numerical experiments and incomplete mathematical statements. In the present paper, we solidify some of the arguments which previously appeared in the literature and offer new insights into the phenomenon.