On rectangular constant in normed linear spaces

TL;DR

This paper investigates the rectangular constant in normed linear spaces, establishing conditions for its maximum value, its attainability in finite dimensions, and characterizations of inner product spaces based on this constant.

Contribution

It provides new characterizations of the rectangular constant and inner product spaces using geometric properties of the unit sphere and the rectangular modulus.

Findings

Rectangular constant equals 3 iff the unit sphere contains a line segment of length 2.

In finite-dimensional spaces, the rectangular constant is attained.

A space is an inner product space iff a specific supremum condition holds for all relevant t values.

Abstract

We study the properties of rectangular constant $ \mu(\mathbb{X}) $ in a normed linear space $\mathbb{X}$. We prove that $ \mu(\mathbb{X}) = 3$ iff the unit sphere contains a straight line segment of length 2. In fact, we prove that the rectangular modulus attains its upper bound iff the unit sphere contains a straight line segment of length 2. We prove that if the dimension of the space $\mathbb{X}$ is finite then $\mu(\mathbb{X})$ is attained. We also prove that a normed linear space is an inner product space iff we have sup$\{\frac{1+|t|}{\|y+tx\|}$: $x,y \in S_{\mathbb{X}}$ with $x\bot_By\} \leq \sqrt{2}$ $\forall t$ satisfying $|t|\in (3-2\sqrt{2},\sqrt{2}+1)$.