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.