On isosceles orthogonality and some geometric constants in a normed space

Abstract

We study the James constant $J(\mathbb{X})$, an important geometric quantity associated with a normed space $\mathbb{X}$, and explore its connection with isosceles orthogonality ${\perp }_I.$ The James constant is defined as $J(\mathbb{X}):=\sup \{\min \{\Vert x+y\Vert ,\Vert x-y\Vert \}:x,y\in \mathbb{X},\Vert x\Vert =\Vert y\Vert =1\}.$ We prove that if $J(\mathbb{X})$ is attained for unit vectors $x,y\in \mathbb{X},$ then $x{\perp }_{I}y.$ We also show that if $\mathbb{X}$ is a two-dimensional polyhedral Banach space then $J(\mathbb{X})$ is always attained at an extreme point $z$ of the unit ball of $\mathbb{X},$ so that $J(\mathbb{X})=\Vert z+y\Vert =\Vert z-y\Vert ,$ where $\Vert y\Vert =1$ and $z{\perp }_{I}y.$ This helps us to explicitly compute the James constant of a two-dimensional polyhedral Banach space in an efficient way. We further study some related problems with reference to several other geometric constants in a normed space.

References in corpus (1)

• The geometry of Minkowski spaces -- a survey. Part I