GEOMETRY OF BISECTORS FOR STRICTLY CONVEX DISTANCES

Abstract

In this paper we study some unexpected geometric properties of the family of bisector lines for a convex distance d, showing that bisectors do not always have an asymptotic line (Section 2). Moreover, although bisectors are homeomorphic to lines, pairs of them can exist intersecting infinitely many times (Section 3). This leads to the conclusion that convex distances are not always nice in the sense of Klein and Wood.7 On the other hand, we prove that distances d, having d-balls whose boundary is given by finitely many algebraic conditions, produce nice distances (Section 3).