= Solution
Assume the standard <empty-coalition normalization> $v(\varnothing)=0$. A <convex cooperative game> has increasing <marginal contributions>: whenever $A\subseteq B$ and $i\notin B$,
$$
v(A\cup\{i\})-v(A)\le v(B\cup\{i\})-v(B).
$$
This follows from the <supermodular> inequality applied to $A\cup\{i\}$ and $B$. Fix a <permutation> $\pi$ of all players, let $P_i$ be the players preceding $i$, and form its <marginal contribution vector>
$$
m_i^\pi=v(P_i\cup\{i\})-v(P_i).
$$
Summing along the full <permutation> is a <telescoping sum>, so $\sum_{i\in N}m_i^\pi=v(N)$. For an arbitrary <coalition> $S$, list its members in their induced order $i_1,\ldots,i_k$, and put $S_j=\{i_1,\ldots,i_j\}$. Since $S_{j-1}\subseteq P_{i_j}$, increasing <marginal contributions> give
$$
\sum_{i\in S}m_i^\pi\ge\sum_{j=1}^k\bigl(v(S_j)-v(S_{j-1})\bigr)=v(S).
$$
Thus every <marginal contribution vector> belongs to the <core>. The vector of <Shapley values> is exactly their uniform average over the $n!$ <permutations>, hence a <convex combination>. By the preceding part, that average is in the <core>. This proves that \b[the <Shapley value> allocation belongs to the <core> of a <convex cooperative game>], including every <coalition> constraint. It is the result <Shapley value belongs to the core of a convex game>.
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