Abstract

In a normed linear space XX an element xx is said to be orthogonal to another element yy in the sense of Birkhoff-James, written as x⊥Byx\perp_{B}y, iff ∥x∥≤∥x+λy∥\|x\| \leq \| x + \lambda y \| for all scalars λ\lambda. We prove that a normed linear space XX is strictly convex iff for any two elements xx, yy of the unit sphere SXS_X, x⊥Byx\perp_{B}y implies ∥x+λy∥>1\|x + \lambda y\| > 1 for all λ≠0\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 XX. Applying the result we give estimations for the lower bounds of ∥tx+(1−t)y∥\|tx+(1-t)y\|, t∈[0,1]t\in [0,1] and ∥y+λx∥\|y + \lambda x\|, for all λ\lambda and for all elements x,y∈SXx,y \in S_X with x⊥Byx\perp_B y. We find a necessary and sufficient condition for the existence of conjugate diameters through the points e1,e2∈SXe_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.

Author information

Contact details are reproduced from the original publication and may be historical.

Kanhaiya Jha

Dept. of Mathematical Sciences, School of Science, Kathmandu University, P.O.Box 6250, Kathmandu, Nepal

Keywords

  • Orthogonality
  • strict convexity
  • extreme point
  • conjugate diameters

Mathematics Subject Classification

46B20; 47A30

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Suggested citation

D. Sain, K. Paul, K. Jha. “Strictly Convex Space: Strong Orthogonality and Conjugate Diameters.” Journal of Convex Analysis 22 (2015), No. 4, 1215–1225. https://doi.org/10.68381/jca22064

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Published by Heldermann Verlag, 2015. Rights now held by Banach Press LLC.