Let $Ax = b$ be a system of hyper-planes that form a bounded convex $D$. Can $D$ be partitioned into union of adjacent simplices?

Question

Let $Ax = b$ be linear system that forms $q$ bounded region $D$. If the columns of $A$ are independent, can $D$ be written as a union of adjacent simplices?

Answer 1

I am did not understand the question completely clear (for instance, it contains no exact definition of adjacent simplexes), but it seems that you are right even without the condition of the column independence of the matrix $A$.

I expect that the required partition of the convex set $D$ can be constructed by induction on the dimension of the faces of $D$. At the beginning of the inductive constuction, for each 1-dimensional face $l$ of $D$ we fix a point $p(l)$ lying in the relative interior of $l$. Denote the set of all such $p(l)$ as $P_1$. At the $k+1$ step of the induction for each $k+1$-dimensional face $l$ of $D$ we fix a point $p(l)$ lying in the relative interior of $l$ and connect $p(l)$ by segment with all points $p(l')\in P_{k}$ such that $l’$ is a $k$-dimensional face of $l$. Denote the set of all such $p(l)$ as $P_{k+1}$. At the last step of the induction, when we consider the point $p(D)$, lying in the relative interior of $D$, we should obtain the partition of $D$ into adjacent simplexes.