Strictly Convex Space: Strong Orthogonality and Conjugate Diameters

In a normed linear space

an element

x x

is said to be orthogonal to another element

y y

in the sense of Birkhoff-James, written as

x\perp_{B}y x ⊥ B y

, iff

\|x\| \leq \| x + \lambda y \| ∥ x ∥ ≤ ∥ x + λ y ∥

for all scalars

\lambda λ

. We prove that a normed linear space

X X

is strictly convex iff for any two elements

x x

,

y y

of the unit sphere

S_X S X

,

x\perp_{B}y x ⊥ B y

implies

\|x + \lambda y\| > 1 ∥ x + λ y ∥ > 1

for all

\lambda \neq 0 λ ≠ 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 X

. Applying the result we give estimations for the lower bounds of

\|tx+(1-t)y\| ∥ t x + ( 1 − t ) y ∥

,

t\in [0,1] t ∈ [ 0 , 1 ]

and

\|y + \lambda x\| ∥ y + λ x ∥

, for all

\lambda λ

and for all elements

x,y \in S_X x , y ∈ S X

with

x\perp_B y x ⊥ 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 e 1 , e 2 ∈ 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.