A short proof for the polyhedrality of the Chvatal-Gomory closure of a compact convex set

Recently Schrijver’s open problem, whether the Chv'atal–Gomory closure of an irrational polytope is polyhedral was answered independently in the affirmative by Dadush, Dey, and Vielma (even for arbitrarily compact convex set) as well as by Dunkel and Schulz. We present a very short, easily accesible proof that the Chv'atal–Gomory closure of a compact convex set is a polytope.

💡 Research Summary

The paper under review presents a remarkably concise proof that the Chvátal‑Gomory (CG) closure of any compact convex set in Euclidean space is a polyhedron. The problem originates from a question posed by Schrijver: whether the CG closure of an irrational polytope remains polyhedral. This question was answered affirmatively in recent work by Dadush, Dey, and Vielma, and independently by Dunkel and Schulz, but their arguments rely on sophisticated tools from measure theory, infinite‑dimensional geometry, and intricate limiting processes. The present contribution strips away that technical baggage and shows that the polyhedrality follows directly from elementary convex analysis combined with the finiteness properties inherent to compact sets.

The authors begin by recalling the definition of the CG closure. For a convex set (K\subseteq\mathbb{R}^n), the CG closure (\operatorname{CG}(K)) is the intersection of all half‑spaces of the form \