Let $T_1$ and $T_2$ be any two euclidean triangles with labeled sides. The sides are labeled respectively $e_1^1,e_2^1,e_3^1$ and $e_1^2,e_2^2,e_3^2$. Call $A:T_1\rightarrow T_2$ the affine map which sends sides to corresponding sides, i.e. $e_i^1\mapsto e_i^2$, $i=1,2,3$.

Call $\mathcal{F}$ the set of differentiable maps which send $T_1$ to $T_2$ and sides to corresponding sides, clearly $A\in \mathcal{F}$. For every $f\in \mathcal{F}$ define $L(f):=\max\{Lip(f),Lip(f^{-1})\}$ where $Lip(f)$ is the Lipschitz constant of $f$.

**Question(s):** Is it true $L(A)=\inf\limits_{f\in\mathcal{F}}L(f)$? If not, is there a map which realizes the infimum? Which one?

Suppose $L(A)=Lip(A)$, then the answer to the first question is yes if $Lip(A)$ is obtained along a side $e$ of $T_1$: $A(e)/e=Lip(A)$, but for every other $f\in\mathcal{F}$ it's true $L(f)\ge f(e)/e\ge A(e)/e=L(A)$.

Unfortunately in case $L(A)$ is not obtained along a side of a triangle the previous inequality can not be used and I don't know how to proceed.

**Additional question:** what are the answers to the previous questions in case one or two sides of $T_1$ are mapped linearly?