Banach Center Publications Volume Institute of Mathematics Polish Academy of Sciences Warszawa Properties of Conflict Sets in the Plane

This paper studies the smoothness and the curvature of con ict sets of the dis tance function in the plane Con ict sets are also well known as bisectors We prove smoothness in the case of two convex sets and give a formula for the curvature We generalize moreover to weighted distance functions the so called Johnson Mehl model Introduction We consider two regions in the Euclidean plane A classical prob lem is to study the set of points of equal distance to the two sets If the sets are points lines or circles we get in this way parabolas ellipses hyperbolas and lines The sets of equal distance are known under several names bisectors in computa tional geometry equi distance lines con ict lines of the distance function in singularity theory cut locus etc Definition Let d denote the Euclidean distance function A and B two closed sets in the plane E By d x A we denote the distance from a point x to A Conf A B fx Ejd x A d x B g Con ict set Terr A B fx Ejd x A d x B g Territory of A w r t B Terr B A fx Ejd x A d x B g Territory of B w r t A We can also start from iso distance lines sets with respect to A and B The con ict sets are precisely the intersections between iso distance lines with the same distance We can imagine the con ict set as the places where wave fronts from A and B meet at the rst time Moreover one can consider circular discs which are tangent to both A and B The centers of the discs are exactly the points on the con ict set If one considers not discs but only circles tangent to both sets one nds in general a bigger locus of centers since then the tangency is also allowed from the inside of the circle The methods of this paper apply with appropriate changes also to this more general type of bisector Mathematics Subject Classi cation Primary A Secondary U The paper is in nal form and no version of it will be published elsewhere