building a polytop from polytop and finding its volume

Question

Let $P$ be a symmetric polytope with $M$ vertices. Suppose we subdivide this polytope into $M$ equal parts $A_i, i=1, \ldots, M$ such that each part $A_i$ correspond to one vertex, $v_i, i=1, \ldots, M$ of the $P$ ( see e.x. Splitting polytope into equal parts).

Consider one part $A_i$. $A_i$ is a symmetric polytope congruent to $P$.

1. How to build up a new polytope $B_i$ from $A_i$ in a way that the vertex $v_i$ would be a centroid of this new polytope $B_i$?

2. What is the volume of $B_i$?