On the Chvátal–Gomory closure of a compact convex set

Abstract

In this paper, we show that the Chvátal–Gomory closure of any compact convex set is a rational polytope. This resolves an open question of Schrijver (Ann Discret Math 9:291–296, 1980) for irrational polytopes, and generalizes the same result for the case of rational polytopes (Schrijver in Ann Discret Math 9:291–296, 1980), rational ellipsoids (Dey and Vielma in IPCO XIV, Lecture Notes in Computer Science, vol 6080. Springer, Berlin, pp 327–340, 2010) and strictly convex bodies (Dadush et al. in Math Oper Res 36:227–239, 2011). An extended abstract of this paper appeared in [6]. After the completion of this work, it has been brought to our notice that the polyhedrality of the Chvátal–Gomory closure for irrational polytopes has recently been shown independently by Dunkel and Schulz [9]. The proof presented in this paper has been obtained independently.