How is the concave closure operation defined? I learned the in a vector space over an ordered field, the convex closure operation of a subset is defined as  the smallest convex set that contains the subset. I was wondering how the concave closure operation is defined?
I also would like to learn about its properties, which in turn might characterize a closure operation to be a concave closure operation. References are also appreciated.
For example, Concave closure of a submodular function
Consider any vector $x=\{x_1,x_2,\dots,x_n\} \in \mathbb{R}^n$ such that each $0\leq x_i\leq 1$. Then the concave closure of a submodular function $f:2^{\Omega}\rightarrow \mathbb{R}$ is defined as
$$f^+(x)=\max(\sum_S \alpha_S f(S):\sum_S \alpha_S 1_S=x,\sum_S \alpha_S=1,\alpha_S\geq 0).$$
What does $1_S$ mean?
What is the range of $S$ in $\sum_S$?
What is the variable $\max$ is taken over?
In fact, I don't understand how the convex closure of a submodular function is related the convex closure of a subset either.
Thanks and regards!
 A: This is not a subject about which I really know anything, but I can at least answer the notational questions.
$S$ ranges over subsets of $\Omega=\{1,2,\dots,n\}$. $1_S$ is the indicator (or characteristic) function of $S$. Or rather, it’s the vector of values of the indicator function, $\langle 1_S(1),1_S(2),\dots,1_S(n)\rangle$. The maximum is taken over all convex combinations $\alpha_S$ of $S\subseteq\Omega$ such that $\sum_S\alpha_S1_S=x$; in a slightly expanded form it’s
$$f^+(\mathbf{x})=\max\left\{\sum_{S\subseteq\Omega}\alpha_Sf(S):\sum_{S\subseteq\Omega}\alpha_S1_S=\mathbf{x}\text{ and }\sum_{S\subseteq\Omega}\alpha_S=1,\text{ where each }\alpha_S\ge0\right\}\;.$$
A: Convex closure here corresponds to the convex closure of a set of points in $\mathbb{R}^{n+1}$ (where $n=|\Omega|$) given by $(1_S,f(S))$ for all subsets $S$ of $\Omega$ ($1_S$ is the indicator vector of S in $\mathbb{R}^n$). 
A dual representation would be in terms of the submodular polyhedron defined as
$$
P_f = \{x \in \mathbb{R}^n: x^T1_S \leq f(S) \forall S \subseteq \Omega\}
$$
Convex closure of $f$ can now be shown to be same as
$$
f^+(c) = \max_{x \in P_f} c^Tx
$$
