Remark in §3.3 — if $X \subseteq [0,1]^n$, every point of $X$ is a vertex of $\mathrm{conv}(X)$

Let $D$ be a rational $m \times n$ matrix and $\mathbf d$ a rational $m$-vector, let $P = \{\mathbf x \in \mathbb R^n \mid D\mathbf x \geqslant \mathbf d,\ \mathbf x \geqslant \mathbf 0\}$ and $X = P \cap \mathbb Z^n$. Suppose $X \subseteq [0,1]^n$, so that $X \subseteq \{0,1\}^n$. Then every point of $X$ is a vertex of the convex hull of $X$:

In this case the discretization (24) and the convexification (23) of the integer program coincide: every integer point is itself one of the extreme points $\mathbf p_q$, so it is the trivial convex combination of one vertex. This is why many set-partitioning and set-covering decompositions need no separate discretization.

Formalization Note "Vertices of conv(X)" is read as the extreme points of the convex hull (Set.extremePoints ℝ (convexHull ℝ X)); since $X$ is finite here, conv(X) is a polytope, whose vertices and extreme points coincide. The statement assumes neither $P \neq \emptyset$ nor anything about $D$ beyond the setting of Theorem 1; the paper states the Remark for the $X$ of Theorem 1.