Extreme concave function on the simplex

Let $$S=\left\{ x\in[0,1]^n : \sum_{i=1}^n x_i=1 \right\}$$ be the $$n$$-simplex. Let $$p\in S$$ be in the interior of $$S$$ (i.e. $$0 for $$i=1,\dots,n$$). Let $$I$$ be the set of extreme points of $$S$$, i.e. $$I=\{ x\in S:\exists i~x_i=1 \}$$. Finally let $$\mathcal H$$ be the set of concave function $$h:S\to\mathbb R$$ such that $$\forall x\in I$$, $$h(x)=0$$ and $$h(p)=1$$.

It is clear that the set $$\mathcal H$$ is convex and so I want to determine its extreme points. It is clear also that the function parametrized by $$q\in S\setminus I$$ and defined as $$h_q:x\to\frac{\min_i\left( \frac{x_i}{q_i}\right)}{\min_i\left(\frac{p_i}{q_i}\right)}$$ is extreme in $$\mathcal H$$. I do not know if these are enough (except for $$n=2$$).

This is pretty much all I have, any idea or literature is very welcome.

Let $$A$$ be a set of $$k\leq n$$ unaligned points in $$S$$. Then the largest concave envelope $$h_A$$ of the set of points $$A\times \{1\} \cup I\times\{ 0\}$$ is concave and in general cannot be expressed as a combination of some $$h_q$$ mentioned above. Also the $$h_q$$ above are of this form (when $$A=\{ q \}$$). Finally it feels like we want some kind of minimality on those sets, for all $$B\subset A$$, the convex hulls of $$A$$ and $$B$$ do not match, i.e. $$\mathcal C(A)\neq\mathcal C(B)$$.

• $0$ is also an extreme point of $S$
– daw
Commented Mar 26, 2021 at 12:32
• Why do you talk about the function value $-\infty$ if you require that $h(x)=0$ for extreme points $x$? This conditions also seems to be violated in your $h_A$ example. It seems like I am misunderstanding something here. Commented Mar 26, 2021 at 13:20
• Right, then I am not interested in the discontinuous case, and I will try to adapt the problem a bit to incorporate it later. Commented Mar 26, 2021 at 13:32