On rectangular constant in normed linear spaces

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$\{\frac{1+|t|}{\Vert y+tx\Vert }$: $x,y\in S_{\mathbb{X}}$ with $x{\perp }_{B}y\}\le \sqrt{2}$ $\forall t$ satisfying $|t|\in (3-2\sqrt{2},\sqrt{2}+1)$.