From what I have seen (and to some extend read), a curve of constant width generated from a polygon with an even number of sides is not possible. Wikipedia cites an Oxford University paper when it says
Curves of constant width can be generated by joining circular arcs centered on the vertices of a regular or irregular convex polygon with an odd number of sides (triangle, pentagon, heptagon, etc.)
It says curves of width can be generated with polygons with an odd number of sides, it does not explicitly rule out even numbers of sides. I suppose this can be asked in two questions:
Question 1 Are all curves of constant width generateable with the "Reaulaux Method"?
It seems that every document I read mentions curves of constant width generated with circular arcs, which does make sense. Is this the only possible method? Examples can be seen on the Wikipedia Article. I'm going to say that a circle can be generated with this method as well. What is the justification for the correct answer?
Question 2 Must a curve of constant width be generated with an odd number of sides?
This question is harder to answer if the answer to question 1 is no, so if that is the case this can be left out (unless it is still easy to answer, that is). Mrf's answer provided insight that could be lead to a proof that regular polygons used to generate with the "Reulaux Method" must have an odd number of sides, but is this also true for irregular polygons? I'm guessing yes but what I need is a proof