Let $T\subset \mathbb{R}^2$ be any triangle and $T^t$ a deformation of $T$. Call $l\_1,l\_2,l\_3$ the squares of the lengths of the sides of $T$ and $l\_1^t,l\_2^t,l\_3^t$ the squares of the lengths of the corresponding sides of $T^t$. Let $\phi\_t:T\rightarrow T^t$ be the affine map which sends sides to corresponding sides.


This is the problem I'd like to solve:


**Find triangles $T$ such that the following equality is verified for any deformation $T^t$:**
$$Max\{\frac{l\_1}{l\_1^t},\frac{l\_2}{l\_2^t},\frac{l\_3}{l\_3^t}\}=Lip(\phi\_t)^2$$


where $Lip(\phi\_t)$ is the Lipschitz constant of $\phi\_t$.



Problem on triangles
The equality is never satisfied.


We send $T$ to $T^t$ by an affine map. This sends the unit circle to an ellipse. The bilipschitz constant is either the half of the major axis or the reciprocal of the half of the minor axis. When we look at how the sides of a triangle are distorted, we "measure the ellipse" in three directions only. There is no guarantee that one of these directions corresponds to the maximum or minimum distortion.


More concretely, for every triangle you can keep the length of one side and move the opposite vertex in a direction perpendicular to that side. The maximum distortion is along this perpendicular and is bigger than the distortion along any of the sides.


On the other hand, one can ask for an estimate of the bilipschitz constant in terms of the distortions along the sides of a triangle...


