Given two convex bodies $A$ and $B$, in $\mathbb R^3$ let's say. We define $A(t)$ and $B(t)$ as $A+xt$ and $B+yt$ where $x,y$ are two arbitrary points. (That is the Minkowski sum, so the two bodies are moving at constant velocity in the $x$ and $y$ directions, and $t$ is the time variable.) Can one show that the function $$f(t)=\operatorname{Vol}(A(t)\cap B(t))$$ is unimodal? That is nondecreasing up to some point, and then nonincreasing.
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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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