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. $0<p_i<1$ 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. $\mathcal C(A)\neq\mathcal C(B)$.

  • $\begingroup$ $0$ is also an extreme point of $S$ $\endgroup$
    – daw
    Commented Mar 26, 2021 at 12:32
  • $\begingroup$ 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. $\endgroup$
    – harfe
    Commented Mar 26, 2021 at 13:20
  • $\begingroup$ Right, then I am not interested in the discontinuous case, and I will try to adapt the problem a bit to incorporate it later. $\endgroup$
    – P. Quinton
    Commented Mar 26, 2021 at 13:32


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