Convex games: equivalence of definitions 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.'.
Define the marginal contribution of $i \in N$ to any $C \subseteq N \setminus \{i\}$ by
\begin{align}
m_i(C) = v(C \cup \{i\}) - v(C). 
\end{align}

We are going to show C.1 $\Rightarrow$ C.2 $\Rightarrow$ C.3 $\Rightarrow$ C.1.
Note that C.1 $\Rightarrow$ C.2 is trivially true.
If C.1 holds for all $S,T \subseteq N$, then C.2 must hold for all $S,T \subseteq N$ with the restriction $|S \setminus T| = |T \setminus S| = 1$.
In order to show C.2 $\Rightarrow$ C.3 consider any $S \subseteq T \subseteq N \setminus \{i\}$.
Let $T \setminus S = \{j_1, \ldots, j_k\}$ and define $S_\ell := T_{\ell-1} \cup \{i\}$ and $T_\ell := T_{\ell-1} \cup \{j_\ell\}$ for $\ell \in \{1,\ldots,k\}$ with $T_0 = S$.
Note that 
\begin{align}
\begin{split}
T_{k-1} &amp;~= T_{k-2} \cup \{j_{k-1}\} \\
&amp;~= T_{k-3} \cup \{j_{k-2}\} \cup \{j_{k-1}\} = T_{k-3} \cup \{j_{k-2},j_{k-1}\}\\
&amp; ~ \vdots\\
&amp;~ =  T_0 \cup \{j_1,\ldots,j_{k-1}\}\\
&amp;~ = S \cup \{j_1,\ldots,j_{k-1}\}\\
&amp;~ = T \setminus \{j_k\}
\end{split}
\end{align}
Since $|S_\ell \setminus T_\ell| = |T_\ell \setminus S_\ell| = 1$ we can apply C.2, i.e.
\begin{align}
\begin{split}
&amp;v(S_\ell \cup T_\ell) + v(S_\ell \cap T_\ell) \geq v(S_\ell) + v(T_\ell)\\
\Longleftrightarrow \quad &amp;v(T_{\ell-1} \cup\{i, j_\ell\}) + v(T_{\ell-1}) \geq v(T_{\ell-1} \cup \{i\}) + v(T_{\ell-1} \cup \{j_\ell\})\\
\Longleftrightarrow \quad &amp;v(T_{\ell-1} \cup\{i, j_\ell\}) - v(T_{\ell-1} \cup \{j_\ell\}) \geq v(T_{\ell-1} \cup \{i\}) - v(T_{\ell-1})\\
\Longleftrightarrow \quad &amp;m_i(T_{\ell-1} \cup\{j_\ell\}) \geq m_i(T_{\ell-1})\\
\Longleftrightarrow \quad &amp;\sum_{\ell=1}^k{m_i(T_{\ell-1} \cup\{j_\ell\})} \geq \sum_{\ell=1}^k{m_i(T_{\ell-1})}\\[2mm]
\Longleftrightarrow \quad &amp;\sum_{\ell=1}^{k-1}{m_i(T_{\ell-1} \cup\{j_\ell\})} + m_i(T_{k-1} \cup\{j_k\}) \geq m_i(T_0) + \sum_{\ell=2}^k{m_i(T_{\ell-1})}\\[2mm]
\Longleftrightarrow \quad &amp;\sum_{\tau=2}^{k}{m_i(T_{\tau-2} \cup\{j_{\tau-1}\})} + m_i((T \setminus \{j_k\}) \cup\{j_k\}) \geq m_i(S) + \sum_{\ell=2}^k{m_i(T_{\ell-1})}\\[2mm]
\Longleftrightarrow \quad &amp;\sum_{\tau=2}^{k}{m_i(T_{\tau-1})} + m_i(T) \geq m_i(S) + \sum_{\ell=2}^k{m_i(T_{\ell-1})}\\
\Longleftrightarrow \quad &amp; m_i(T) \geq m_i(S)
\end{split}
\label{eq:pvex}
\end{align}
The bottom line corresponds to C.3.
The final step C.3 $\Rightarrow$ C.1 is provided in Moulin (1988, p. 113).