2 enh detail

The sets $\{ (A(t),t)|t\in \mathbb{R} \}$ } \subset \mathbb{R}^4$and$\{ (B(t),t)|t\in \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.

1

The sets $\{ (A(t),t)|t\in \mathbb{R} \}$ and $\{ (B(t),t)|t\in \mathbb{R} \}$ 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.