A conjectured class of scale-invariant distances on inner product spaces

Abstract: Let $V$ be an inner product space, and $x, y \in V$; the conjecture is made that, for any $p \in [1, \infty]$, the function $d_p(x, y):=|x-y|/(|x|^p+ |y|^p)^{1/p}$ is a distance on $V$.