2016/01/18 by J. Jerónimo-Castro, Jerónimo-Castro, J., E. Makai Jr +1
Mathematics · #52A55 #FOS: Mathematics #Metric Geometry (math.MG) #math.MG #msc:52A55
paper · pdf · doi:10.48550/arxiv.1601.04494
44 pdf file pages
arxiv created 2016/01/18 · arxiv updated 2016/01/19
High proved the following theorem. If the intersections of any two congruent copies of a plane convex body are centrally symmetric, then this body is a circle. In our paper we extend the theorem of High to spherical and hyperbolic planes. If in any of these planes, or in \Bbb R2, there is a pair of closed convex sets with interior points, and the intersections of any congruent copies of these sets are centrally symmetric, then, under some mild hypotheses, our sets are congruent circles, or, for \Bbb R2, two parallel strips. We prove the analogue of this statement, for Sd, \Bbb Rd, Hd, if we suppose C2+: again, our sets are congruent balls. In S2, \Bbb R2 and H2 we investigate a variant of this question: supposing that the numbers of connected components of the boundaries of both sets are finite, we exactly describe all pairs of such closed convex sets, with interior points, whose any congruent copies have an intersection with axial symmetry (there are 1, 5 or 9 cases, respectively).