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June 2018 Geometric Aspects of $p$-angular and Skew $p$-angular Distances
Somayeh HABIBZADEH, Mohammad Sal MOSLEHIAN, Jamal ROOIN
Tokyo J. Math. 41(1): 253-272 (June 2018). DOI: 10.3836/tjm/1502179269

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 $\alpha_p[x,y]\geq\beta_p[x,y]$.

Citation

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Somayeh HABIBZADEH. Mohammad Sal MOSLEHIAN. Jamal ROOIN. "Geometric Aspects of $p$-angular and Skew $p$-angular Distances." Tokyo J. Math. 41 (1) 253 - 272, June 2018. https://doi.org/10.3836/tjm/1502179269

Information

Published: June 2018
First available in Project Euclid: 26 January 2018

zbMATH: 06966868
MathSciNet: MR3830818
Digital Object Identifier: 10.3836/tjm/1502179269

Subjects:
Primary: 46B20
Secondary: 26D15, 46C15

Rights: Copyright © 2018 Publication Committee for the Tokyo Journal of Mathematics

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Vol.41 • No. 1 • June 2018
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