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, andRead more
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