Co-convexial reflector curves with applications (Q1089123)

In solving a problem related to the Stieltjes and Van Vleck polynomials [J. Math. Anal. Appl. 110, 327-339 (1985; Zbl 0589.30004)], the authors conjectured that every convex compact subset of the complex plane is of reflecting type. In this paper, they answer their conjecture in the affirmative by the introduction of the so-called co-convexial reflector curves. Moreover, they identify some interesting geometrical features by showing that this family of reflector curves supports a theory analogous to that of confocal ellipses. As applications, they obtain new results on the zeros of Stieltjes and Van Vleck polynomials, some of which were predicted in the reference.