On disjunction convex hulls by big-M lifting

We study the natural extended-variable formulation for the disjunction of $n+1$ polytopes in $R^d$. We demonstrate that the convex hull $D$ in the natural extended-variable space $R^{d+n}$ is given by full optimal big-M lifting (i) when $d\le 2$ (and that it is not generally true for $d\ge 3$), and also (ii) under some technical conditions, when the polytopes have a common facet-describing constraint matrix, for arbitrary $d\ge 1$ and $n\ge 1$. We give a broad family of examples with $d\ge 3$ and $n=1$, where the convex hull is not described after employing all full optimal big-M lifting inequalities, but it is described after one round of MIR inequalities. Additionally, we give some general results on the polyhedral structure of $D$, and we demonstrate that all facets of $D$ can be enumerated in polynomial time when $d$ is fixed.