Motzkin decomposition of closed convex sets (Q847066)

It is well known that a polyhedral set (i.e., the intersection of a finite number of closed half-spaces) can be expressed as the (Minkowski) sum of a polytope and a polyhedral cone. In the paper, the authors present extensions of this result. They give necessary and sufficient conditions such that an arbitrary closed convex set \(F\subset\mathbb{R}^n\) is Motzkin representable, i.e., representable as the sum of a convex compact set and a closed convex cone. The approach is closely connected with other types of representations of a closed convex set: the linear representation of \(F\) as the solution set of a semi-infinite (possibly small) system of linear inequalities according to \(\{x\in\mathbb{R}^n\mid a^\top_t x\geq b_t, t\in T\}\) and the conic representation of \(F\) using the polar cone and the bipolar cone of the larger set \(\text{cone}(F\times\{-1\})\subset\mathbb{R}^{n+1}\).