= Solution
\b[True, with the usual normalization $v(\varnothing)=0$.] For a <convex cooperative game>, the <supermodular> inequality implies increasing <marginal contributions>: if $A\subseteq B$ and $i\notin B$, apply it to $A\cup\{i\}$ and $B$ to obtain
$$
v(A\cup\{i\})-v(A)\le v(B\cup\{i\})-v(B).
$$
Fix an ordering $\pi$ and let $P_i$ be the set of players before $i$. Its <marginal contribution vector> is $m_i^\pi=v(P_i\cup\{i\})-v(P_i)$. Summing in order telescopes to $\sum_i m_i^\pi=v(N)$. For any <coalition> $S$, $S\cap P_i\subseteq P_i$, so increasing marginals give
$$
\sum_{i\in S}m_i^\pi\ge\sum_{i\in S}\bigl(v((S\cap P_i)\cup\{i\})-v(S\cap P_i)\bigr)=v(S).
$$
These are exactly the efficiency and <coalition> constraints of the <core of a cooperative game>. Thus every <marginal contribution> vector is in the <core>. The <core> is a <convex set>, being an intersection of linear <half-spaces> and an efficiency <hyperplane>. The <Shapley value> is the average of the <marginal contribution> vectors over all orderings, so it too lies in the <core>. This proves <Shapley value belongs to the core of a convex game>, without needing a separate existence theorem for the <core>.
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