Let $X$ be a finite subset of $\mathbb R$ and let $f : X \to {\mathbb R}$. Suppose we want to approximate $f$ by a polynomial $g$ of fixed degree $d\geq 1$ with the additional condition $g\geq f$. Let

$$ S=\lbrace h : X \to {\mathbb R} | h \geq f, \ h {\rm\ is \ the \ restriction \ of \ a \ polynomial \ } g {\rm \ of \ degree \ } d\rbrace $$

We equip $S$ with the usual partial ordering ( $h_1 \leq h_2$ iff $h_1(x) \leq h_2(x)$ for all $x\in X$). Let $M(S)$ denote the set of minimal elements of $S$. Is it true that $M(S)$is always finite ? Also, is there an effective bound for $|M(S)|$ in terms of $d$ ?

When $d=1$, it seems that $|M(S)| \leq 3$ (this bound is attained when the graph of the function is a trapezium for example).

UPDATE : as noted in the comment below, $M(S)$ is not finite. But $S$ is convex, so the convex hull $C(M(S))$ of $M(S)$ is contained in $S$, and it seems that the set of extremal points in $C(M(S))$ (denote it by $E(S)$) is finite.