This is an announcement for the paper "On rectangular constant in normed linear spaces" by Kallol Paul, Puja Ghosh, and Debmalya Sain.
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 $|t|\in (3-2\sqrt{2},\sqrt{2}+1)$.
Archive classification: math.FA
Mathematics Subject Classification: 46B20, 47A30
Submitted from: kalloldada@gmail.com
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1407.1353
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