Strictly convex space : Strong orthogonality and Conjugate diameters

Abstract

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 $\Vert x\Vert \le \Vert x+\lambda y\Vert$ 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 $\Vert x+\lambda y\Vert >1\forall \lambda \ne 0.$ We apply this result to find a necessary and sufficient condition for a Hamel basis to be a strongly orthonormal Hamel basis in the sense of Birkhoff-James in a finite dimensional real strictly convex space X. Applying the result we give an estimation for lower bounds of $\Vert tx+(1-t)y\Vert ,t\in [0,1]$ and $\Vert y+\lambda x\Vert ,\forall \lambda$ 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 conjuagte diameters is then developed for a real strictly convex smooth space of finite dimension.