A New Algorithm for Approximating the Least Concave Majorant

The least concave majorant, $\hat F$, of a continuous function $F$ on a closed interval, $I$, is defined by $$\hat F (x) = \inf \left\{ G(x): G \geq F, G \mbox{ concave}\right\},\; x \in I.$$ We present here an algorithm, in the spirit of the Jarvis March, to approximate the least concave majorant of a differentiable piecewise polynomial function of degree at most three on $I$. Given any function $F \in \mathcal{C}^4(I)$, it can be well-approximated on $I$ by a clamped cubic spline $S$. We show that $\hat S$ is then a good approximation to $\hat F$. We give two examples, one to illustrate, the other to apply our algorithm.