The critical point and related symmetry measures of a planar convex set
Given any point \(x\) on the interior of a compact convex set, and a line through that point, we have \(x=\lambda y+(1- \lambda)z\) where \(y\) and \(z\) are the unique two points at which the boundary of the set; and this uniquely determines \(\lambda\in (0,1)\). By compactness, \(\lambda\) achieves a minimum value \(\lambda(x)\) as the line varies through all possible lines through \(x\); and there is a unique critical point \(x^*\) that maximizes \(\lambda(x)\). The maximum value, \(\lambda(x^*)\), is called the critical value \(\mu^*\); it is an affine invariant. We may think of \(x^*\) as the point that comes closest to being a center of symmetry, and \(\mu^*\) as a measure of how close the body comes to central symmetry. Of course, central symmetry can be defined in other ways. For instance, for a centrally symmetric body, the distance between a hyperplane through a point and one parallel support is always half that between the two parallel supports. Minimizing this ratio over all hyperplanes through a point, and maximizing over all points, we obtain a different affine invariant \(\mu\), but the point at which the maximum occurs is the same. Applying the same ``maximin'' procedure to the proportion of the body's volume that is on one side of the hyperplane yields a different affine invariant \(\nu\) and a different critical point. Again, if we consider the proportion of the body's surface measure that is on one side of the hyperplane, we obtain yet another critical point; this time, the associated measure of symmetry \(\xi\) is not an affine invariant. Upper and lower bounds for these measures of symmetry are known, but the lower bound for \(\xi\) is not sharp. A conjecture of Neumann suggests a lower bound that is approached in the limit of thin isosceles triangles. In this paper, the author gives several algorithms for computing the various critical points and symmetry measures of a convex polygon. Examples and graphs are given, which clarify the exposition. It is claimed that the computations support Neumann's conjecture, although this is not clear from the examples given, which do not appear to be particularly extreme cases.
scientific article; zbMATH DE number 978120