Farthest points on most Alexandrov surfaces

Abstract

We study global maxima of distance functions on most Alexandrov surfaces with curvature bounded below, where most is used in the sense of Baire categories.Math. Subj. Classification (2010): 53C45

Farthest points on flat surfaces

“…In [9] we extend Tudor Zamfirescu's results in [17], [18], and show that on most (in the sense of Baire categories) Alexandrov surfaces outside the aforementioned components, most points have a unique farthest point. The cases of the connected components of flat surfaces, namely the components of A(0) containing flat tori and flat Klein bottles respectively, are treated in this note with elementary techniques.…”

“…A motivation for this note lies in the application of Baire Categories theorem to the study of farthest points on Alexandrov surfaces, initiated in [15] and continued in [9]. Roughly speaking, an Alexandrov surface with curvature bounded below by κ is a 2-dimensional topological manifold endowed with an intrinsic metric which verifies Toponogov's comparison property.…”