Cross-posted from math.stackexchange:

Let C be a simple closed plane curve and let D be its interior. Recall that the width of C in a direction θ is the distance between two supporting lines for D which are perpendicular to θ. A curve is said to have constant width if its width is the same in every direction; see the Reuleaux polygons for nontrivial examples.

It is often stated (for instance in "The Enjoyment of Math" by Rademacher and Toeplitz) that curves of constant width are necessarily convex. Does anyone know how to prove this? I can't find any references, and a proof eludes me.