Geometric aspects of $p$-angular and skew $p$-angular distances

Abstract

Corresponding to the concept of $p$-angular distance ${\alpha }_p[x,y]:=\Vert \Vert x{\Vert }^{p-1}x-\Vert y{\Vert }^{p-1}y\Vert$, we first introduce the notion of skew $p$-angular distance ${\beta }_p[x,y]:=\Vert \Vert y{\Vert }^{p-1}x-\Vert x{\Vert }^{p-1}y\Vert$ for non-zero elements of $x,y$ in a real normed linear space and study some of significant geometric properties of the $p$-angular and the skew $p$-angular distances. We then give some results comparing two different $p$-angular distances with each other. Finally, we present some characterizations of inner product spaces related to the $p$-angular and the skew $p$-angular distances. In particular, we show that if $p>1$ is a real number, then a real normed space $\mathcal{X}$ is an inner product space, if and only if for any $x,y\in \mathcal{X}\setminus \{0\}$, it holds that ${\alpha }_p[x,y]\ge {\beta }_p[x,y]$.

20 pages, to appear in Tokyo J. Math