# Rolling/width functions: Characterization?

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?

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.

• You might have a relation between the smoothness of the body boundary and the smoothness of the resulting curve. Just playing around with parts of polygons, it looks like you can make an impressive variety of curves. Maybe you can find appropriate results in linkages ("most manifolds can be well approximated by the state space of a linkage") that would apply here. Gerhard "Is Often Looking For Link(age)s" Paseman, 2018.01.20. – Gerhard Paseman Jan 20 '18 at 21:53
• I think you should withdraw (or more clearly indicate support for) the statement about f(x) consisting of two congruent pieces. The issue I see is one of speed: the side containing a may be rotating faster than the side containing b when you draw f(x). There may be some congruency in a quantity derived from f, but your first animation seems to belie your last statement. Gerhard "Belie Means False? Not True?" Paseman, 2018.01.22. – Gerhard Paseman Jan 22 '18 at 16:16
• @GerhardPaseman: Oh you are right---Thanks! – Joseph O'Rourke Jan 22 '18 at 17:43
• Instead of rolling the body along the straight line, roll the line along the boundary of the body. For each point $p$ on the boundary, $f(x(p)) = h(u(p)) + h(-u(p))$, where $u(p)$ is the unit normal at $p$, $h$ is the support function, and $x$ is arclength. The normal $u$ is the integral of curvature with respect to $x$. – Deane Yang Jan 23 '18 at 21:00