Let us assume that we are given a set $S=\{x\in \mathbb{R}^n:h(x)\le 0\}$, and we are given a set of points $p_1,\cdots,\ p_{n+1}$, such that $p_i\in S$. My question is:
Is there a systematic way of checking whether the set $S$ belongs to the convex hull of the set of points $\{p_1,\cdots,\ p_{n+1}\}$?
I tried using contradictions, that is I assumed that there is a point $v\in S$, such that $v$ cannot be represented as a convex combination of the points $p_1,\cdots,p_{n+1}$, and tried to get a contradiction out of it by showing that $v$ will satisfy $h(v)>0$. However, the success of this approach seems to be highly dependent on the specific $h$ and the points $p_1,\cdots,\ p_{n+1}$. Can someone provide me with some reference to relevant literature where such cases are discussed and analyzed in some generality, i.e., results are derived for some general class of functions $h$?
Thanks in advance.