On isosceles orthogonality and some geometric constants in a normed space

TL;DR

This paper investigates the James constant in normed spaces, its relation to isosceles orthogonality, and provides methods to compute it explicitly in two-dimensional polyhedral Banach spaces.

Contribution

It establishes conditions under which the James constant is attained and connects it with isosceles orthogonality, enabling explicit computation in specific spaces.

Findings

James constant is attained at orthogonal vectors.

Explicit computation of James constant in 2D polyhedral spaces.

Connection between James constant attainment and isosceles orthogonality.

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 $J(\mathbb{X}) := \sup\{\min \{\|x+y\|, \|x-y\|\}: x, y \in \mathbb{X},~ \|x\|=\|y\|=1 \}.$ 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 $J(\mathbb{X}) = \|z+y\| = \|z-y\|,$ where $ \| y \| = 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.