On the covering property of the lifting region

We will present recent progress on the connections functions and covering/tiling properties of subsets of euclidean sets. Important structural information about strong cut-generating functions can be translated to geometric questions like: Does a particular compact subset X of R^n cover all of R^n when we consider all of its translates by integer vectors? This connects to very classical problems in the geometry of numbers and deep theorems like the Venkov-Alexandrov-McMullen theorem on tilings, and the geometry of zonotopes can be leveraged. Research in this area of integer optimization is very much work-in-progress; we will close the presentation with an invitation to join our quest with some open problems.