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Kyungpook Mathematical Journal 2023; 63(2): 199-223

Published online June 30, 2023 https://doi.org/10.5666/KMJ.2023.63.2.199

Copyright © Kyungpook Mathematical Journal.

On Some Skew Constants in Banach Spaces

Yuankang Fu, Zhijian Yang and Yongjin Li∗, Qi Liu

Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275, P. R. China
e-mail : fuyk5@mail2.sysu.edu.cn, yangzhj55@mail2.sysu.edu.cn and stslyj@mail.sysu.edu.cn

School of Mathematics and Physics, Anqing Normal University, Anqing, 246133, P. R. China
e-mail : liuq67@aqnu.edu.cn

Received: May 2, 2022; Revised: February 10, 2023; Accepted: March 14, 2023

Abstract

We introduce the constants E[t,X], CNJ[X] and J[t,X] to describe the asymmetry of the norm. They can be seen as the skew version of the Gao's parameter, von Neumann-Jordan constant and Milman's moduli, respectively. We establish basic properties of these constants, relating them other well known constants, and use these properties to calculate the constants for specific spaces. We then use these constants to study Hilbert spaces, uniformly non-square spaces and their normal structures. With the Banach-–Mazur distance, we use them to study isomorphic Banach spaces.

Keywords: geometric constant, uniform non-squareness, normal structure