DOI: 10.68381/jca24054 ISSN: 0944-6532
On Rectangular Constant in Normed Linear Spaces
Kallol Paul, Puja Ghosh, Debmalya Sain
We study the properties of rectangular constant
\mu(\mathbb{X})
μ
(
X
)
in a normed linear space
\mathbb{X}
X
. We prove that
\mu(\mathbb{X}) = 3
μ
(
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}
X
is finite then
\mu(\mathbb{X})
μ
(
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.