This is an announcement for the paper "Strictly convex space : Strong orthogonality and conjugate diameters" by Debmalya Sain, Kallol Paul and Kanhaiya Jha.
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 $ | 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~ \forall~ \lambda \neq 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 $ | tx+(1-t)y|, t \in [0,1] $ and $ | y + \lambda x |, ~\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.
Archive classification: math.FA
Mathematics Subject Classification: Primary 46B20, Secondary 47A30
Submitted from: kalloldada@gmail.com
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1411.1464
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