Is it overkill to invoke Kirszbraun theorem to prove the following fact ? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T22:32:42Z http://mathoverflow.net/feeds/question/89866 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/89866/is-it-overkill-to-invoke-kirszbraun-theorem-to-prove-the-following-fact Is it overkill to invoke Kirszbraun theorem to prove the following fact ? Thomas Richard 2012-02-29T13:25:38Z 2012-02-29T13:25:38Z <p>Given a small enough convex triangle $(abc)$ in a (smooth or Alexandrov) surface $(X,d)$ of curvature greater than $-1$, let $(\overline{abc})$ be its comparison triangle in $\mathbb{H}^2$. Then there exist a $1$-Lipschitz map $\varphi$ from the interior of $(abc)$ to the interior of $(\overline{abc})$.</p> <p>Using Kirszbraun theorem (for Alexandrov spaces, as stated by Lang and Schroeder or Alexander, Kapovitch and Petrunin) this is simple. Just identify the sides of $(abc)$ and $(\overline{abc})$ in the usual way, this map is $1$-Lipschitz because of the curvature bound and we can apply Kirszbraun theorem to extend it to the interior of the triangle.</p> <p>I was wondering if one could explicitly build $\varphi$ in this specific case.</p>