  Consider a bounded open set $U$ in $R^n$ and a Lipschitz function g from the
boundary of $U$ to $R^m$. Does this function always have a canonical optimal
Lipschitz extension to all of $U$? We propose a notion of optimal Lipschitz
extension and address existence and uniqueness in some special cases. In the
case $n=m=2$, we show that smooth solutions have two phases: in one they are
conformal and in the other they are variants of infinity harmonic functions
called infinity harmonic fans. We also prove existence and uniqueness for the
extension problem on finite graphs.
