Denote by $\mathbb{N}$ the set of positive integers. Denote by $\mathbb{R}$ the set of real numbers. Denote by $\mathbb{R}_+$ the set of non-negative real numbers. For every pair of numbers $i, j \in \mathbb{N}$ we define $[i..j] := \{k \in \mathbb{N}\ |\!:\ i \leq k\leq j\}$.
Fix an arbitrary $n \in \mathbb{N}$.
For every pair of real-valued $n$-tuples $\mathbf{a} = (a_1, \dots, a_n), \mathbf{b} = (b_1, \dots, b_n)\in\mathbb{R}^n$ we write $\mathbf{a} \geq \mathbf{b}$ iff $a_i \geq b_i$ for every $i \in [1..n]$.
A function $f:A\subseteq\mathbb{R}^n\rightarrow\mathbb{R}$ shall be called (1) monotonic iff for every $\mathbf{a}, \mathbf{b} \in A$ such that $\mathbf{a} \geq \mathbf{b}$, we have $f(\mathbf{a}) \geq f(\mathbf{b})$, (2) positive-homogeneous iff $f(c\mathbf{a}) = cf(\mathbf{a})$ whenever $c \in \mathbf{R}_+$, and $\mathbf{a}, c\mathbf{a} \in A$.
Denote by $I_n$ the unit square in $\mathbb{R}^n$, i.e. $I_n := [0,1]^n$. Denote by $\mathbf{1}_n$ the $n$-tuple all of whose components are $1$: $\mathbf{1}_n := (\underbrace{1, \dots, 1}_n)$. Observe that $\mathbf{1}_n \in I_n$.
Denote by $V_n$ the standard real vector space on $\mathbb{R}^n$. Denote by $\left<\cdot, \cdot\right>$ the standard inner product associated with $V_n$. In other words, for every pair of real-valued $n$-tuples $\mathbf{a} = (a_1, \dots, a_n), \mathbf{b} = (b_1, \dots, b_n) \in \mathbb{R}^n$, we have $\left<\mathbf{a}, \mathbf{b}\right> = \sum_{i = 1}^n a_i b_i$.
For every function $v:A\subseteq I_n\rightarrow\mathbb{R}$, denote by $\mathcal{L}_v$ the set consisting of all $V_n$-concave, monotonic, and positive-homogeneous functions $f:\mathbb{R}_+^n\rightarrow\mathbb{R}$ such that $f(\mathbf{a}) \geq v(\mathbf{a})$ for every $\mathbf{a} \in A$.
I wish to show the following. Let $\mathcal{P} \subseteq \mathbb{R}_+^n$ be a non-empty, finite set of $n$-tuples over $\mathbb{R}_+$. There exists a finite subset of the unit square $A \subseteq I_n$ with $\mathbf{1}_n \in A$, and a monotonic function $v:A\rightarrow\mathbb{R}_+$, such that for every $\mathbf{b} \in \mathbb{R}_+^n$ we have $$ \inf_{f \in \mathcal{L}_v} f(\mathbf{b}) = \min_{\mathbf{p} \in \mathcal{P}} \left<\mathbf{b}, \mathbf{p}\right>. $$
This above is a self-contained restatement of a part of an unproved proposition that appears in the following paper (Proposition 9.29(1) on p. 117). According to the author (ibid.),
The proof is rather standard and is therefore omitted.
The original proposition consists of two claims. My post describes a part of the first of these claims. The following hint, given in the paper (on p. 117), seems to pertain to the second claim, but it may also be helpful for the part that I am interested in, so I will quote it here:
It [the proof] is based on the fact that any concave function over a compact and convex set $D$, that can be extended as a concave function to an open set that contains $D$, is the minimum of all its supporting linear functions.