This post is centred on planar, compact, convex sets of constant width. It updates part of my March 2010 post here. For terminology and basic results see H.G. Eggleston, âConvexityâ, CUP, 1969 et al. .
That earlier post contained another statement of the following problem. Let p be the common circumcentre and incentre of a (compact, convex) set X of constant width . Let f(x,p) be the foot of the perpendicular from p to a support hyperplane to X at x: x a frontier point of X. If there is more than one support hyperplane at x let the support hyperplane be chosen to maximise the f/g ratio below. Let g(x,p) be the unique frontier point of X between p and f(x,p). Now consider the ratio of the distance of f(x,p) from p and the distance of g(x,p) from p and call this the f/g ratio.
The earlier post also contained the conjecture that for planar X the ratio was maximised for a Reuleaux triangle. The conjectured maximum value should have been 1.093 not 1.116. An easy upper bound comes from the ratio of circumradius to inradius: see Theorem 53 of âConvexityâ.
While trying to understand planar sets of constant width, helped by the wonderful Mathematica 8 software and literature searches, I began to wonder whether all such constant width extremal problems have the disc or Reuleaux triangle as extremal sets. Of course, approximation and convergence arguments suggest Reuleaux polygons, of constant curvature except for finitely many points, are central.
A simple measure of asymmetry, for sets of constant width 1(say) , when the origin is located at the common incentre and circumcentre, is the maximum value of the support function which it is natural to conjecture is maximised for Reuleaux triangles.
Fujiwaraâs 1927 proof of the Blaschke-Lebesgue theorem contains an important reference to a 1917 paper with Kakeya: both available free via the Journal @rchive website. A cursory study suggests that the Japanese school of mathematics in the early twentieth century seems to have explored such questions as the problems above extensively?
Helpful comments on these questions, from anyone, would be appreciated.