mg.metric geometry - Pushing convex bodies together

The sets ${ (A(t),t)|tin mathbb{R} } subset mathbb{R}^4$ and ${ (B(t),t)|tin mathbb{R} } subset mathbb{R}^4$ are convex, their intersection $K$ is a bounded convex set, and $f(t)$ is the volume of the slice of $K$ at height $t$. By Brunn-Minkowski inequality, this is log-concave, so definitely unimodal.