User ondra - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T16:22:54Z http://mathoverflow.net/feeds/user/4302 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/16681/is-ellipse-on-a-sphere-convex-proof Is ellipse on a sphere convex? (proof) ondra 2010-02-28T14:58:04Z 2010-03-01T04:39:07Z <p>Is 'small enough' ellipse projected on a surface of a sphere convex? By ellipse I mean a set of points 'C' with a constant sum |AC| + |BC|, A and B are the centers. By 'small enough' I mean that the radii fits into 90 degrees (I think it is not convex once you make it large enough, though the limit is probably more like 180 degrees).</p> <p>It seems to me that it is indeed convex, but is there some simple proof? The mathematics I tried to do usually ends up as f(x)=arccos(a(x)) + arccos(b(x)) and it isn't quite easy to prove that that the function has a reasonable shape when a(x) is decreasing and b(x) is increasing. Is there some easy proof I have overlooked?</p> <p>By convex I mean that any shortest line connecting the points on the ellipse is 'inside' the ellipse (i.e. the distance |AX| + |XB| is smaller or equal then the distance defining the ellipse for any point X of the line).</p> <p>Update: I think I eventually found a solution; triangle inequality works for these 'small enough' triangles. Geometrically the problem can be somewhat shuffled, in the end I have to prove that a triangle that is 'inside' another triangle is indeed smaller; triangle inequality combined with the way of computing a distance on a sphere will do the trick.</p> http://mathoverflow.net/questions/16681/is-ellipse-on-a-sphere-convex-proof/16710#16710 Comment by ondra ondra 2010-02-28T18:41:16Z 2010-02-28T18:41:16Z Yep, that's roughly the solution I later found out.