Abstract of a paper by Kallol Paul, Puja Ghosh, and Debmalya Sain
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 or http://arXiv.org/abs/1407.1353
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alspach@math.okstate.edu