Let $K$ be a strictly convex planar body of perimeter $1$. Roll it along the $x$-axis from $0$ to $1$, and define $f(x)$ to be the height of the highest point of $K$ when it touches at $(x,0)$. So $f(x)$ is the width between supporting parallel lines.

. Is there some characterization of the functions realizable in this manner?Q

Or is there not much more that can be said beyond: the width functions of a convex body?

One obvious constraint rules out a function like this:

If that minimum width is realized by parallel lines supporting $K$ at boundary points $a$ and $b$, then it would reappear with the roles of $a$ and $b$ reversed. So every $f(x)$ value should appear at least twice.