19. (POL 1) 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 $T$.
19. Let $d_{1}, d_{2}, d_{3}, d_{4}$ be the distances of the point $P$ to the tetrahedron. Let $d$ be the height of the regular tetrahedron. Let $x_{i}=d / d$. Clearly, $x_{1}+$ $x_{2}+x_{3}+x_{4}=1$, and given this condition, the parameters vary freely as we vary $P$ within the tetrahedron. The four tetrahedra have volumes $x_{1}^{3}, x_{2}^{3}, x_{3}^{3}$, and $x_{4}^{3}$, and the four parallelepipeds have volumes of $6 x_{2} x_{3} x_{4}$, $6 x_{1} x_{3} x_{4}, 6 x_{1} x_{2} x_{4}$, and $6 x_{1} x_{2} x_{3}$. Hence, using $x_{1}+x_{2}+x_{3}+x_{4}=1$ and setting $g(x)=x^{2}(1-x)$, we directly verify that
$$
\begin{aligned}
f(P) & =f\left(x_{1}, x_{2}, x_{3}, x_{4}\right)=1-\sum_{i=1}^{4} x_{i}^{3}-6 \sum_{1 \leq k0$ we have $g\left(x_{i}+x_{j}\right)+g(0) \geq$ $g\left(x_{i}\right)+g\left(x_{j}\right)$. Equality holds only when $x_{i}+x_{j}=2 / 3$.
Assuming without loss of generality $x_{1} \geq x_{2} \geq x_{3} \geq x_{4}$, we have $g\left(x_{1}\right)+$ $g\left(x_{2}\right)+g\left(x_{3}\right)+g\left(x_{4}\right)<g\left(x_{1}\right)+g\left(x_{2}\right)+g\left(x_{3}+x_{4}\right)$. Assuming $y_{1}+y_{2}+y_{3}=1$ and $y_{1} \geq y_{2} \geq y_{3}$, we have $g\left(y_{1}\right)+g\left(y_{2}\right)+g\left(y_{3}\right) \leq g\left(y_{1}\right)+g\left(y_{2}+y_{3}\right)$. Hence $g\left(x_{1}\right)+g\left(x_{2}\right)+g\left(x_{3}\right)+g\left(x_{4}\right)<g(x)+g(1-x)$ for some $x$. We also have $g(x)+g(1-x)=x(1-x) \leq 1 / 4$. Hence $f(P) \leq 3 / 4$. Equality holds for $x_{1}=x_{2}=1 / 2, x_{3}=x_{4}=0$ (corresponding to the midpoint of an edge), and as the variables converge to these values, $f(P)$ converges to $3 / 4$. Hence the bounds for $f(P)$ are
$$
0<f(P)<\frac{3}{4}
$$