Geodesic
On Rectangular Constant in Normed Linear Spaces
Journal of convex analysis, Tome 24 (2017) no. 3, pp. 917-925
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We study the properties of rectangular constant $\mu(\mathbb{X})$ in a normed linear space $\mathbb{X}$. We prove that $\mu(\mathbb{X}) = 3$ if and only if the unit sphere contains a straight line segment of length 2. In fact, we prove that the rectangular modulus attains its upper bound if and only if 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 find a necessary and sufficient condition for a normed linear space to be an inner product space in terms of conditions involving rectangular constant.
Classification : 46B20, 47A30
Mots-clés : Birkhoff-James Orthogonality, rectangular constant 
@article{JCA_2017_24_3_JCA_2017_24_3_a9,
     author = {K. Paul and P. Ghosh and D. Sain},
     title = {On {Rectangular} {Constant} in {Normed} {Linear} {Spaces}},
     journal = {Journal of convex analysis},
     pages = {917--925},
     year = {2017},
     volume = {24},
     number = {3},
     url = {http://geodesic.mathdoc.fr/item/JCA_2017_24_3_JCA_2017_24_3_a9/}
}
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K. Paul; P. Ghosh; D. Sain. On Rectangular Constant in Normed Linear Spaces. Journal of convex analysis, Tome 24 (2017) no. 3, pp. 917-925. http://geodesic.mathdoc.fr/item/JCA_2017_24_3_JCA_2017_24_3_a9/