Let $N \subset \mathbb N$ denote the set of players and $v : 2^N \to \mathbb R$ , $v(\emptyset) = 0$, the characteristic function. We call $(N,v)$ a cooperative transferable utility (TU) game.
Definition 1. A game $(N,v)$ is called convex, if for all $S,T \subseteq N$ it holds that \begin{align} v(S \cup T) + v(S \cap T) \geq v(S) + v(T) \tag{C.1}. \end{align} In Hafalir (2007, p. 255) it says that Def. (1) is equivalent to the following statement.
Definition 2.
A game $(N,v)$ is called convex, if for all $S,T \subseteq N$ with $|S \setminus T| = |T \setminus S| \leq 1$ it holds that
\begin{align}
v(S \cup T) + v(S \cap T) \geq v(S) + v(T) \tag{C.2}.
\end{align}
There is no proof of the statement, but the author refers to Moulin (1988, p. 112). But here it only gives an equivalent for Definition 1.
Definition 1'.
A game $(N,v)$ is called convex, if for all $i \in N$ and $S \subset T \subset N \setminus \{i\}$ it holds that
\begin{align}
v(S \cup \{i\}) - v(S) \leq v(T \cup \{i\}) - v(T) \tag{C.3}.
\end{align}
Definitions 1. and 1'. are standard. I never heard of 2. though. I have a problem, however, where I can show that Def. 2. is satisfied. And now I'm wondering how to show the equivalence of Def. 2. with either 1. or 1.'.