linear algebra – The convex hull of a vector space.

For a matrix $A = (a_1, ldots, a_n) in mathbb{R}^{m times n}$, denote
begin{equation*}
| A|_{b, c} := big| (| a_1|_b, ldots, | a_n|_b)^{top} big|_c.
end{equation*}

Now suppose $V subseteq mathbb{R}^n$ and $A in mathbb{R}^{n times m}$, does the following relationship hold?
begin{equation*}
big{ A^{top} b : b in V big} = big{ | A^{top} |_{1, infty} b : b in operatorname{conv} (-V cup V)big},
end{equation*}

where $operatorname{conv}$ is the convex hull of a set.