Theorem 4.7 — $K_2$ is closed and convex, and polyhedral when $\operatorname{pos}(\tilde\Xi_{p,T})$ is a polyhedral cone

Let $W$ be a fixed $\bar m \times \bar n$ matrix, $\mu$ the (probability) law of $\xi = (c, q, p, T)$, $\tilde\Xi_{p,T}$ the support of the marginal law of $(p,T)$, and

Then:

1. $K_2$ is a closed convex subset of $\mathbb{R}^n$;
2. if the closed positive hull $\operatorname{pos}(\tilde\Xi_{p,T})$ (the closure of the set of nonnegative linear combinations of points of $\tilde\Xi_{p,T}$) is a convex polyhedral cone, then $K_2$ is a convex polyhedron, i.e. $K_2 = \{x \mid Gx \ge \alpha\}$ for some finite system of linear inequalities.

No moment condition on $\xi$ is needed. Part 1 says the induced feasibility region is always well behaved; part 2 gives a checkable condition, on the distribution of $(p, T)$ alone, under which it is described by finitely many linear constraints.

Formalization Note. A convex polyhedral cone is taken to be the set of nonnegative combinations of finitely many vectors of $(p,T)$-space $\mathbb{R}^{\bar m} \times \mathbb{R}^{\bar m \times n}$; a convex polyhedron is the solution set of finitely many weak linear inequalities (the empty set and $\mathbb{R}^n$ included). The number of inequalities is an existentially quantified natural number.