2015/06/07 by Rouyer, Joël
#51A15 #52B10 #53C45 #FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.1506.02284
On a convex surface S, the antipodal map F associates to a point p the set of farthest points from p, with respect to the intrinsic metric. S is called a Steinhaus surface if F is a single-valued involution. We prove that any convex polyhedron has an open and dense set of points p admitting a unique antipode Fp, which in turn admits a unique antipode FFp, distinct from p. In particular, no convex polyhedron is Steinhaus.