On 2/23/07, Daniel Asimov <dasimov@earthlink.net> wrote:

There ought to be a simple criterion under which there exists a unique "best" simple closed curve in the plane.

E.g.g., suppose

[Groan!]

1) The curve has bilateral symmetry;

2) the curvature attains two local maxima with values k = M and k = m;

Visual inspection of the proposals so far suggests that a good egg has (at least) 6 vertices rather than 4, with a curvature max at the sharp end and min at the blunt end. [The "flat spots" in RWG's example, recently villified elsewhere in this --- inexplicably fragmenting --- thread, emphasise the other two minima --- and how!] I don't think this alters your proposal in any essential fashion. However ...

3) the interior of the curve has a fixed area A.

Then I'd think there might be a unique "optimal" such curve having minimum length

(or equally, fix the length and maximize the area),

and that the unique optimal curve would automatically be analytic.

(If so, then quite possibly condition 1) is redundant.)

Let's take another leaf from the proof in the paper by Gluck et al which you mentioned earlier, and approximate to this curve by circular arcs. The result is easily seen to be a circle for almost its entire perimeter, with small pimples (or flats) at just two places --- whose locations, by the way, are arbitrary. So the limit is essentially a circle, but for two singular points! An alternative attack might be to specify the max and min curvatures, and their distance along the bilateral axis, then minimise the maximum absolute value of d(curvature)/d(arc length) = d^2(tangent angle)/d(arc length)^2 over the curve. Even here, I suspect the result may be discontinuous in its third derivative at the vertices. This kind of hassle seems inevitable in any construction which attempts simply to specify the precise behaviour of a curve --- open or closed --- at some point in configuration space. Witness the complicated behaviour of any exact interpolation method with the flexible simplicity of Bernstein/ Bezier/ deCasteljau/ NURBS splines. Fred Lunnon