TITLE: Orthogonality in non-inner product spaces
QUESTION [10 upvotes]: I have come across a notion of orthogonality of two vectors in a normed space not necessarily inner product space. Two vectors $x$ and $y$ in a normed space are said to be orthogonal (represented $x\perp y$) if $||x||\leq ||x+\alpha y||,$ for every $\alpha,$ a scalar. 
1) What is the rational behind the definition above? I guess, it has got something to do with minimum overlap between $x$ and $y$. 
2) Is this unique generalization of the concept of orthogonality from inner product spaces?
Thank you.

REPLY [4 votes]: Concerning your follow-up question (iii) there is the following very nice result: For Birkhoff-James orthogonality 
it is easy to find examples where $y\perp x$ but 
$\left\|x\right\|/\left\|x+\alpha y\right\| > 1$ for some real $\alpha$, and so 
natural to investigate the largest such value 
$\left\|x\right\|/\left\|x+\alpha y\right\|$ over $X$. 
In "R. L. Thele, Some results on the radial projection in Banach spaces.
Proc. Amer. Math. Soc., 42(2):484--486", it is it is shown that this quantity
is exactly the Lipshitz constant for the radial projection onto the unit ball
in this norm.