Show that a function is well-defined

Question

The following is the lemma $2$.

I don't understand the last sentence. Which calculations give $L(x)=0$? And why this explains the well-definedness of the function $L$?

Answer 1

In this case you have $$\|p_q(x)\| = \max_{s\ne q} \|x_s\| = \|x_r\| = \|x_q\|.$$ Hence, $$1-\frac{\|p_q(x)\|}{\|x_q\|} = 0$$ and similarly for $r$.