Let us say that two norms $\|\cdot\|_1$ and $\|\cdot\|_2$ on a real vector space $V$ are *strongly equivalent* if there exists a constant $\lambda \geq 1$ such that 
$$
\frac{1}{\lambda} \left( \|x\|_1 + \|y\|_1 - \|x + y\|_1 \right) \leq 
 \|x\|_2 + \|y\|_2 - \|x + y\|_2 \leq  \lambda \left( \|x\|_1 + \|y\|_1 - \|x + y\|_1 \right)
$$
for all vectors $x$ and $y$ in $V$.

A remark I owe to Suvrit is that if we take $y = -x/2$, this condition implies that 
$$
\frac{1}{\lambda} \|x\|_1 \leq  \|x\|_2 \leq  \lambda \|x\|_1
$$
so that the norms are equivalent in the usual sense.

Geometrically, two norms are strongly equivalent if the defect in the triangular inequality for any one of the norms is controlled by the defect of the other. In particular, both normed spaces must have *exactly* the same geodesics. Thus, for example, the $\ell_\infty$ and the $\ell_2$ norms on the plane are **not** strongly equivalent. 

**Question.** Is there a simple criterion to determine whether two norms on the plane (or in a finite-dimensional space) are strongly equivalent. What if we assume that the unit spheres of both norms are polygons or that the unit spheres are smooth and positively curved?