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

## TL;DR

This paper introduces and studies the geometric properties of $p$-angular and skew $p$-angular distances in normed spaces, providing characterizations of inner product spaces based on inequalities involving these distances.

## Contribution

It defines the skew $p$-angular distance, explores its properties, compares it with the $p$-angular distance, and characterizes inner product spaces through these distances.

## Key findings

- The skew $p$-angular distance is introduced and analyzed.
- Comparison results between $p$-angular and skew $p$-angular distances are established.
- A characterization of inner product spaces is provided based on inequalities involving these distances.

## Abstract

Corresponding to the concept of $p$-angular distance $\alpha_p[x,y]:=\left\lVert\lVert x\rVert^{p-1}x-\lVert y\rVert^{p-1}y\right\rVert$, we first introduce the notion of skew $p$-angular distance $\beta_p[x,y]:=\left\lVert \lVert y\rVert^{p-1}x-\lVert x\rVert^{p-1}y\right\rVert$ 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 $x,y\in \mathcal{X}\smallsetminus{\lbrace 0\rbrace}$, it holds that $\alpha_p[x,y]\geq\beta_p[x,y]$.