Over the past few years, interest has increased in models defined on positive and negative integers. Several application areas lead to data that are differences between Some important examples price changes measured discretely financial applications, pre- posttreatment measurements of discrete outcomes clinical trials, difference number goals sports events, differencing count-valued time series...

Motivated by recent advances in solution methods for mixed-integer convex optimization (MICP), we study the fundamental and open question of which sets can be represented exactly as feasible regions MICP problems. We establish several results this direction, including first complete characterization mixed-binary case a simple necessary condition general case. use latter to derive non-representa...