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 $d_p(x,y):=\Vert x-y\Vert /(\Vert x{\Vert }^p+\Vert y{\Vert }^p{)}^{1/p}$ is a distance on $V$.

arXiv commentThis paper has been withdrawn by the author. The conjecture was already been made and solved. See Peter A. Hästö, A new weighted metric: the relative metric I, J. Math. Anal. Appl. 274, 38-58 (2002). Thanks to prof. Heinz Bauschke