Points joined by three shortest paths on convex surfaces

Abstract

Let<italic>S</italic>be a convex surface and<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x element-of upper S"><mml:semantics><mml:mrow><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:mi>S</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">x \in S</mml:annotation></mml:semantics></mml:math></inline-formula>. It is shown here that the set of all points of<italic>S</italic>joined with<italic>x</italic>by at least three shortest paths can be dense in<italic>S</italic>. It is proven that, in fact, in the sense of Baire categories most convex surfaces have this property, for any<italic>x</italic>. Moreover, on most convex surfaces, for most of their points, there is just one farthest point (in the intrinsic metric), and precisely three shortest paths lead to that point.