On the Chvatal-Gomory Closure of a Compact Convex Set

TL;DR

This paper proves that the Chvatal-Gomory closure of any compact convex set is a rational polytope, resolving a longstanding open question and extending previous results to irrational polytopes.

Contribution

It establishes that the CG closure of a compact convex set is always a rational polytope, generalizing prior results for specific convex sets.

Findings

CG closure of a compact convex set is a rational polytope

Resolves an open question from Schrijver 1980

Extends known results to irrational polytopes

Abstract

In this paper, we show that the Chvatal-Gomory closure of a compact convex set is a rational polytope. This resolves an open question discussed in Schrijver [Schrijver 80'] and generalizes the same result for the case of rational polytopes [Schrijver 80'], rational ellipsoids [Dey-Vielma 10'] and strictly convex sets [Dadush-Dey-Vielma 10']. In particular, it shows that the CG closure of an irrational polytope is a rational polytope, which was the open question in [Schrijver 80'].