Newsgroups:sci.mathSubject:bifocals?From:elkies@ramanujan.harvard.edu (Noam Elkies)Date:20 Sep 92 15:12:26 EDTOrganization:Harvard Math DepartmentSummary:does a bifocal curve exist?

Say that a point P in the interior of a simple closed curve C in the plane is a _focal point_ of C if C is starlike w.r.t. P and all the chords of C passing through P have the same length. Is it known whether a simple closed curve can have more than one focal point? [C is starlike w.r.t. P means every ray from P intersects C in exactly one point; this is a condition weaker than convexity (though C is convex iff it is starlike w.r.t. every interior point). There are plenty of curves with one focal point.] --Noam D. Elkies (elkies@zariski.harvard.edu) Dept. of Mathematics, Harvard University

From:ctm@math.berkeley.edu (Curtis T. McMullen)Newsgroups:sci.mathSubject:Re: bifocals?Date:20 Sep 1992 20:21:39 GMTOrganization:U.C. Berkeley Math. Department.

Let a *wheel* be a Jordan curve W in the plane. A chord for W is a line segment with endpoints on W. A point p in the plane is an *axle* for W if there exists an L>0 such that for every w in W, there is a w' in W such that |w-w'| = L and the segment [w,w'] contains p in its interior. Then a less restrictive form of Elkies' question is: does there exist a wheel with two axles? Here are some recollection of my thoughts on this problem from 1990 or so. It can be related to dynamical systems as follows. Let p and q be two (candidate) axles; I think one can show they both must have the same L, by considering the line through p and q. Given a point w, allegedly on the wheel, one can construct two more points w_1 and w_2 on the wheel by drawing lines of length L from w through p and w through q. Repeating this process, you can make a computer drawing of (lots of) points which must lie on the wheel W to be consistent with the data (w,p,q,L). By starting with points w very close to the line through p and q, you can generate continua which must lie in W. These continua tend to be non-locally connected, due to incoherence between stable and unstable manifolds in this dynamical system (a well-known phenomena). I suspect this can be made into a proof that there is no wheel with two axes. It should be considerably simpler with the assumption that the wheel is star-shaped from p and q. But perhaps the discussion is "academic", since a solution of a large portion of the problem appears in: @article{Schafke:Volkmer, key="SV", author={R. Sch\"afke and H. Volkmer}, title="Asymptotic analysis of the equichordal problem", journal="J. reine angew. Math.", volume="425",year="1992",pages="9-60"} -Curt McMullen

From:orourke@whatever.cs.jhu.edu (Joseph O'Rourke)Newsgroups:sci.mathSubject:Equichordal problemDate:9 Jan 91 17:36:34 GMTReply-To:orourke@cs.jhu.edu (Joseph O'Rourke)Organization:Smith College, Northampton MA USA

A year ago I heard that the equichordal problem had at long last been solved, but I have not seen the paper. Does anyone have a reference? The equichordal problem is to determine whether a compact convex subset S of E^2 might have more than one equichordal point. A point p is equichordal in S if every chord through p has the same length.

To:geometry-research@forum.swarthmore.eduDate:2 Oct 1997 21:55:41 -0400From:Marek Rychlik <rychlik@math.arizona.edu>Organization:EpigoneSubject:Re: Equichordal points

The best known outstanding problem regarding equichordal points has just recently been solved by me (solution published in Inventiones Mathematicae, 129(1), pp. 141-212, 1997). This problem was posed in 1916 by Fujiwara and in 1917 by Blaschke et. al. : A point P inside a closed convex curve C is called equichordal if every chord of C drawn through point P has the same length. Is there a curve with two equichordal points? It proves that a curve cannot have two equichordal points. The assumption of convexity can be significantly weakened, to include all Jordan curves. For further references my Web site can be consulted: http://alamos.math.arizona.edu -Marek Rychlik