>On Jul 18, 2:43Â pm, Mark Steinberger <nyj...@gmail.com> wrote: >> Does anyone know an elementary proof that great circles in the sphere >> minimize arc length? > >There is a triangle inequality in spherical geometry as well (as long >as you consider points on the same half-sphere, which is always >possible). If we are allowed to assume this, then it becomes very >simple. > >Also, proving this is not so bad.
One ought also to prove that the metric for which the triangle inequality has been assumed or proved (presumably the metric such that the distance between P and Q is the product of the radius of the sphere and the central angle between P and Q) is identical to the metric generated by arc length (i.e., such that the distance between P and Q is the infimum of the arc lengths of rectifiable curves joining P to Q), while being careful not to make a (not great...) circular argument.
Taking such care, however, is likely to cause "elementary"-ness to fly out the window. But since it's Mark's question, he gets to decide on what "elementary" means in it. So far he has not responded (publicly) to any of the proposed answers.