From: ikastan@alumnae.caltech.edu (Ilias Kastanas)
Newsgroups: sci.math
Subject: Re: CLOSED CURVE problem.
Date: 6 Apr 1996 13:38:54 GMT

In article <4jvacb$ghc@cantua.canterbury.ac.nz>,
Bill Taylor <mathwft@math.canterbury.ac.nz> wrote:
>This question is well-known, I'm sure, and correctly answered in the negative,
>it would seem, but I'm ****ed if I can see why.
>
>--------------
>Consider the class of convex, closed curves in R^2, sufficently smooth enough
>to have many continuous derivatives.
>
>Is it possible for any to have a single point of locally maximum curvature?
>--------------
>
>It seems not, but why not?  There's no simple topological reason against it;
>at least in that the curvature expressed as a function of circular [0,1) could
>have one local maximum, in principle.   What stops this?


	The "accumulation of small advantages" does, a chess player would say.

	Vogt's theorem:  If a curve has nondecreasing curvature (of constant
   sign) from point P to point Q, and q = angle of the tangent vector at Q to
   chord PQ, p = same at P, then  q >= p. (q = p only if curvature is constant)

	An integration by parts is enough to prove this.  Or, if you keep
   making that vector turn faster, it will.

	In the case at hand, if there were only one point of maximum (minimum)
   curvature, we could take it as Q (P).  The two arcs from P to Q yield
   q1 > p1 and q2 > p2.   But  p1 + p2 = pi = q1 + q2.


	So there are at least four "vertices", two maxima and two minima.


							Ilias
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Recently came across this entry in MathSciNet: --- djr

   97e:53006 53A04 51L15
   Martinez-Maure, Yves
   A note on the tennis ball theorem. (English)
   Amer. Math. Monthly 103 (1996), no. 4, 338--340.

   The tennis ball theorem of the title is due to V. I. Arnold [
   Topological invariants of plane curves and caustics, Amer. Math. Soc.,
   Providence, RI, 1994; MR 95h:57003] and states that a closed simple
   smooth curve in the 2-sphere which divides the sphere into two parts
   with equal areas has at least 4 inflexions. (An inflexion here is a
   zero of geodesic curvature: the curve projected out onto the tangent
   plane to the sphere has an inflexion.) 

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