
Journal of Convex Analysis 22 (2015), No. 4, 12151225 Copyright Heldermann Verlag 2015 Strictly Convex Space: Strong Orthogonality and Conjugate Diameters Debmalya Sain Dept. of Mathematics, Jadavpur University, Kolkata 700032, India saindebmalya@gmail.com Kallol Paul Dept. of Mathematics, Jadavpur University, Kolkata 700032, India kalloldada@gmail.com Kanhaiya Jha Dept. of Mathematical Sciences, School of Science, Kathmandu University, P.O.Box 6250, Kathmandu, Nepal [Abstractpdf] In a normed linear space $X$ an element $x$ is said to be orthogonal to another element $y$ in the sense of BirkhoffJames, written as $x\perp_{B}y$, iff $\x\ \leq \ x + \lambda y \$ for all scalars $\lambda$. We prove that a normed linear space $X$ is strictly convex iff for any two elements $x$, $y$ of the unit sphere $S_X$, $x\perp_{B}y$ implies $\x + \lambda y\ > 1$ for all $\lambda \neq 0$. We apply this result to find a necessary and sufficient condition for a Hamel basis to be strongly orthonormal in the sense of BirkhoffJames in a finite dimensional real strictly convex space $X$. Applying the result we give estimations for the lower bounds of $\tx+(1t)y\$, $t\in [0,1]$ and $\y + \lambda x\$, for all $\lambda$ and for all elements $x,y \in S_X$ with $x\perp_B y$. We find a necessary and sufficient condition for the existence of conjugate diameters through the points $e_1,e_2 \in S_X$ in a real strictly convex space of dimension 2. The concept of generalized conjugate diameters is then developed for a real strictly convex smooth space of finite dimension. Keywords: Orthogonality, strict convexity, extreme point, conjugate diameters. MSC: 46B20; 47A30 [ Fulltextpdf (128 KB)] for subscribers only. 