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Extremal Sets of Constant Width and the Japanese School
Posted:
Feb 29, 2012 9:30 AM
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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.
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