## Extermum points of the intersection between simplex of probability and hyperplanes

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 $A\cap D =\operatorname{conv}(q_1,\ldots,q_r)$.

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$D$ is the same as $\Delta$ I presume. The intersection of a $r$ dimensional linear space and a simplex is certainly a convex polytope, but it can well have more than $r$ vertices. – Pietro Majer Oct 7 2011 at 9:11