= Integer restrictions on a convex feasible region
If the amounts in the budget example must be <integers>, the actual <feasible region> is $C\cap\mathbb Z^2$. Both $(0,0)$ and $(1,0)$ are feasible, but their <midpoint> $(1/2,0)$ is not. Thus the continuous <feasible region> is a convex set while the integer-restricted choices are not. Averaging two manufacturing plans may specify half an indivisible item. <Integer programming> must respect that restriction; using the surrounding convex set is a relaxation, not a proof that its averages are admissible integer plans.
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