Let $S^n = \{x \in \mathbb{R}^{n+1}; \langle\, x,x\rangle = 1 \}$.
$P^n$ is the set of all unordered pairs $[x] = \{x,-x\}$, $x \in S^n$.
I'd like to prove that $d([x],[y]) = \min \{\|x-y\|,\|x+y\|\}$ is a metric. I'm having trouble with the triangle inequality.
I looked at this question and I found the answer didn't clarify the situation for me.
It should be simple, but I didn't understand the steps 2 and 3 in the answer. Why is it possible to choose such representation of $x$ and $y$?
How about this? \begin{align*} d([x],[z])&=\min(||x-z||,||x+z||)\\ &\leq \min(||x-z||+||y-z||,||x+z||+||y+z||)\\ &\leq \min(||x-z||,||x+z||)+\min(||y-z||,||y+z||) \end{align*} Where the second line is true because we add a positive number to positive numbers and as such the minimum must be greater and the last one is fairly self evident.