Width Deviation of Convex Polygons

We consider the width XT (ω) of a convex n-gon T in the plane along the random direction ω ∈ R/2πZ and study its deviation rate: δ(XT ) = √ E(X2 T )− E(XT )2 E(XT ) . We prove that the maximum is attained if and only if T degenerates to a 2-gon. Let n ≥ 2 be an integer which is not a power of 2. We show that √ π 4n tan( π 2n ) + π2 8n2 sin( π 2n ) − 1 is the minimum of δ(XT ) among all n-gons and determine completely the shapes of T ’s which attain this minimum. They are characterized as polygonal approximations of equi-Reuleaux bodies, found and studied by K. Reinhardt [9]. In particular, if n is odd, then the regular n-gon is one of the minimum shapes. When n is even, we see that regular n-gon is far from optimal. We also observe an unexpected property of the deviation rate on the truncation of the regular triangle.