Given integers $k < n$, let
$$ \mathcal{V} := \left\{ {\bf x} \in \{0, 1 \}^n \mid {\bf 1}_n^\top {\bf x} = k \right\} $$
Is it true that its convex hull is $\mathcal{S} := \left\{ {\bf x} \in[0,1]^n \mid {\bf 1}_n^\top {\bf x} = k \right\}$?
Because, it holds for $n>1$ and $k=1$, i.e., the unit simplex is the convex hull of the standard basis. But when I think about $k>1$, things seem to go crazy. Any good ideas or good references would be greatly appreciated!