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.
Q. Is there some characterization of the functions realizable in this manner?
Or is there not much more that can be said beyond: the width functions of a convex body?
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