Convex conjugate of a constrained quadratic ($({\chi }_{[-1,1]}+\frac{1}{2}|\cdot {|}^2{)}^{\ast }(p)=\{\begin{array}{ll}{p}^2/2,& |p|\le 1,\\ |p|-1/2,& |p|>1.\end{array}$)

Maximize $px-x^2/2$ over $|x|\le 1$. 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.