Let
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
be a point inside a regular tetrahedron
![T](/media/m/0/1/6/016d42c58f7f5f06bdf8af6b85141914.png)
of unit volume. The four planes passing through
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
and parallel to the faces of
![T](/media/m/0/1/6/016d42c58f7f5f06bdf8af6b85141914.png)
partition
![T](/media/m/0/1/6/016d42c58f7f5f06bdf8af6b85141914.png)
into 14 pieces. Let
![f(P)](/media/m/9/6/e/96ef4ffb9f4c8901e36bda14cf22aa13.png)
be the joint volume of those pieces that are neither a tetrahedron nor a parallelepiped (i.e., pieces adjacent to an edge but not to a vertex). Find the exact bounds for
![f(P)](/media/m/9/6/e/96ef4ffb9f4c8901e36bda14cf22aa13.png)
as
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
varies over
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Let $P$ be a point inside a regular tetrahedron $T$ of unit volume. The four planes passing through $P$ and parallel to the faces of $T$ partition $T$ into 14 pieces. Let $f(P)$ be the joint volume of those pieces that are neither a tetrahedron nor a parallelepiped (i.e., pieces adjacent to an edge but not to a vertex). Find the exact bounds for $f(P)$ as $P$ varies over $T.$