Question

Let $\Delta$ be a probability simplex in $\mathbb{R}^d$. Now let $p_1,\ldots,p_n$ be a set of probability vectors in $\Delta$ where $\operatorname{Rank} \operatorname{span}({p_1,\ldots,p_n})=r$. Let $A$ be the set of all possible linear (including but not limited to convex) combinations of $p_1,\ldots,p_n$. Can it be shown that the intersection of $A$ and $D$ is an $r-1$-dimensional polytope. In other word, do there exist linearly independent $q_1,\ldots, q_r \in D$ such that $A\cap D =\operatorname{conv}(q_1,\ldots,q_r)$.