The following inequality is an elementary exercise in convexity: let $x,y$ be non-zero vectors in a normed space with $\|x\|, \|y\|\leqslant 1$. Suppose that $\|x-y\| \geqslant 1$. Then
$$\left\|\frac{x}{\|x\|} - \frac{y}{\|y\|} \right\| \geqslant \|x-y\|.$$
It is stated in Lemma 6 of
H. Martini, K. J. Swanepoel, and G. Weiß, The geometry of Minkowski spaces – a survey. Part I, Expositiones Math. 19 (2001), 97–142.
I was wondering if anyone knows the original reference (or, at least an older one)?
