On some arc length properties in Minkowski planes

Abstract

In this note we discuss a property in Minkowski planes related to the (intrinsic) arc length. Let A and B be two distinct non diametral points on the unit sphere, H the midpoint of the chord AB and C = H/∥H∥ : we say that for the (shortest) arc joining A and B the arc length property holds if the greatest length of the two arcs joining respectively A, C and C, B is smaller than half the length of the chord plus the norm of the segment CH. We discuss cases in which the arc length property holds and cases in which fails, in particular we show that this property holds for “small” arcs when the sphere is twice differentiable. Preliminary to the proof of this last result we have established some formulas on arc parametrized twice differentiable spheres, here the main tools used are James orthogonality and the duality mapping: as a further application a “short” proof of Schäffer’s theorem on the perimeter of spheres is given.

Keywords: Minkowski planes, arc length, arc parametrization.