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.
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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. |
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