Let $\Delta$ be a probability simplex in $R^d$. \mathbb{R}^d$. Now let $p_1,\ldots,p_n$ be a set of probability vectors in $\Delta$ where $Rank(span{p_1,\ldots,p_n})=r$. \operatorname{Rank} \operatorname{span}({p_1,\ldots,p_n})=r$. Let A $A$ 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 $A$ and D $D$ is an $r-1$-dimensional polytope. In other word, do there exist linearly independent $q_1,\ldots, q_r \in D$ such that $A\cap D =conv(q_1,\ldots,q_r)$.\operatorname{conv}(q_1,\ldots,q_r)$.
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Extermum points of the intersection between simplex of probability and hyperplanesLet $\Delta$ be a probability simplex in $R^d$. Now let $p_1,\ldots,p_n$ be a set of probability vectors in $\Delta$ where $Rank(span{p_1,\ldots,p_n})=r$. 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 $A\cap D =conv(q_1,\ldots,q_r)$.
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