Box and triangle intersection
The following answer is incorrect.

Suppose the triangle has vertices $(x_1,y_1,z_1), (x_2,y_2,z_2), (x_3,y_3,z_3)$ and the box has coordinates $[x_\min, x_\max] \times [y_\min,y_\max] \times [z_\min, z_\max]$.  Then the bounding box of their intersection is $[x'_\min, x'_\max] \times [y'_\min,y'_\max] \times [z'_\min, z'_\max]$, where
\begin{align*}
    x'_\min &amp;= \max\{\min\{x_1,x_2,x_3\}, x_\min\}, \\
    x'_\max &amp;= \min\{\max\{x_1,x_2,x_3\}, x_\max\}, \\[1ex]
    y'_\min &amp;= \max\{\min\{y_1,y_2,y_3\}, y_\min\}, \\
    y'_\max &amp;= \min\{\max\{y_1,y_2,y_3\}, y_\max\}, \\[1ex]
    z'_\min &amp;= \max\{\min\{z_1,z_2,z_3\}, z_\min\}, \\
    z'_\max &amp;= \min\{\max\{z_1,z_2,z_3\}, z_\max\}. \\[1ex]
\end{align*}