  We give a simple procedure to estimate the smallest Lipshitz constant of a
degree 1 map from a Riemannian 2-sphere to the unit 2-sphere, up to a factor of
10. Using this procedure, we are able to prove several inequalities involving
this Lipshitz constant. For instance, if the smallest Lipshitz constant is at
least 1, then the Riemannian 2-sphere has Uryson 1-Width less than 12 and
contains a closed geodesic of length less than 160. Similarly, if a closed
oriented Riemannian surface does not admit a degree 1 map to the unit 2-sphere
with Lipshitz constant 1, then it contains a closed homologically non-trivial
curve of length less than 4 pi. On the other hand, we give examples of high
genus surfaces with arbitrarily large Uryson 1-Width which do not admit a map
of non-zero degree to the unit sphere with Lipshitz constant 1.
