Convex conjugate of a constrained quadratic
= Convex conjugate of a constrained quadratic
{title2=$\bigl(\chi_{[-1,1]}+\tfrac12|\cdot|^2\bigr)^*(p)=\begin{cases}p^2/2,&|p|\leq1,\\|p|-1/2,&|p|>1.\end{cases}$}
Maximize $px-x^2/2$ over $|x|\leq1$. The maximizer is $x=\max(-1,\min(p,1))$, giving the displayed <Huber loss>. The <indicator functional of a constraint set> is zero on the interval and infinity outside; it is not the zero-one indicator used in <measure theory>.