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.



How do you prove that every curve of constant width is convex?
First, observe that for a closed convex region $R$ of constant width $w$, there can be no interval $[a,b]\subset \partial R$. For take the support line $l$ to $R$ containing $[a,b]\subset l$, and let $c$ be a point in the other parallel support line for $R$ realizing the width $w$. Then the height of the triangle $abc$ is $w = d(c,l)$. However, the distance $w<\max\{d(a,c),d(b,c)\}$, and therefore the width of $R$ will be greater that the maximum of these two distances by choosing the support lines perpendicular to the longer side, contradicting constant width. 


Now, suppose $K$ is a constant width region, and let $R$ be the convex hull of $K$. If $K \neq R$, then there exists an interval $[a,b]\subset \partial R-\partial K$, a contradiction. 


