I gathered different handouts from the internet and tried to figure out what fenchel transformation was, but I stumbled on two different results of fenchel transformation of $f(x) = |x|$:
First results is:
Second result is:
I believe the first one is correct but can anyone confirm this for me? In addition, should there be vertical lines at $p=-1$ and $p=1$?
The result is written explicitly in the wikipedia page about convex conjugate. In your case, one has $$ f^\star(x^*) = \sup_\limits{x\in \mathbb{R}}\{x^* x - |x|\} =\left\{{0\text{ if } |x^*|\le 1\atop +\infty\text{ if }|x^*|\gt 1}\right. $$
Your first picture tries to represent the subderivative, a related concept.
Edit: the biconjugate is here
$$ f^{\star\star}(x) = \sup_\limits{x^*\in\mathbb{R}}(x^* x - f^\star(x^*)) = \sup_\limits{x^*\in[-1, 1]}(x^* x) = |x| = f(x) $$
This is the expected result because $f$ is convex.