Strictly Convex Space: Strong Orthogonality and Conjugate Diameters

In a normed linear space $X$ an element $x$ is said to be orthogonal to another element $y$ in the sense of Birkhoff-James, 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 Birkhoff-James in a finite dimensional real strictly convex space $X$. Applying the result we give estimations for the lower bounds of $\|tx+(1-t)y\|$, $t\in [0,1]$ 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.