Solution
= Solution
For the <absolute value function>, $px-|x|\leq0$ for all $x$ if $|p|\leq1$, with equality at $x=0$. If $|p|>1$, taking $x=t\operatorname{sign}(p)$ with $t\to\infty$ makes the expression diverge. Thus its <convex conjugate> is
$$
\boxed{E^*(p)=\begin{cases}0,&|p|\leq1,\\+\infty,&|p|>1.\end{cases}}
$$
This is the <indicator functional of a constraint set> for the dual unit interval, not a zero-one indicator function.