Let $S$ be a finite set, and let $W$ be a nonempty set of subsets of $S$; we will identify every subset of $S$ with its characteristic function, a 0-1 vector in $\mathbb R^S$. The combinatorial dual $\hat W$ of $W$ is the set of all subsets of $S$ that have non-empty intersection with every subset of $W$. The positive linear dual, for our purposes, is the set $W^\perp$ of all $v \in \mathbb R^S$ with non-negative entries such that $v \cdot w \ge 1$ for all $w \in W$. The convex production of a finite set $F \subset [0, \infty)^S$ is the set of vectors of the form $v + w$, where $v$ is in the convex hull of $F$, and $w \in [0, \infty)^S$. My first question is
Is $W^\perp$ always the convex production of $\hat W$?
It is clear that the latter is a subset of the former.
My second question is
Is the distance the origin of $W^\perp$ equal to the distance to the origin of the convex hull of $\hat W$?
Of course an affirmative answer to the first question would imply an affirmative answer to the second.
This question is motivated by a related question in the theory of combinatorial extremal length.