= Solution
Write $x(S)=\sum_{i\in S}x_i$. The <core of a cooperative game> is
$$
C(v)=\{x\in\mathbb R^n:x(N)=v(N),\quad x(S)\ge v(S)\text{ for every }S\subseteq N\}.
$$
Take any $x,y\in C(v)$ and $\alpha\in[0,1]$, and form the <convex combination> $z=\alpha x+(1-\alpha)y$. Its total <payoff> is $z(N)=\alpha v(N)+(1-\alpha)v(N)=v(N)$. For every <coalition> $S$,
$$
z(S)=\alpha x(S)+(1-\alpha)y(S)\ge\alpha v(S)+(1-\alpha)v(S)=v(S).
$$
Hence $z$ remains in the <core>. \b[The <core> is a <convex set>], even when the <transferable utility game> is not a <convex cooperative game>. An empty <core> is a <convex set> vacuously; the argument does not claim nonemptiness. Geometrically, the same conclusion follows because the constraints intersect an <affine hyperplane> with <closed half-spaces>.
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