# Problem on triangles

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 $\phi_t$.

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.
• It works in the first order case as well, although I agree that it is vague. The concrete example I gave should leave no doubts: when the square of the height $h_1$ is increased by $t$, the squares of $l_2$ and $l_3$ are also increased by $t$. Since $l_2, l_3 > h_1$ (we assume there is no right angle at the side $l_1$), the first-order relative distortions of $l_2$ and $l_3$ are smaller than that of $h_1$. – Ivan Izmestiev Nov 17 '16 at 14:45