If a norm $\lVert \cdot \rVert$ is strictly convex and $\lambda \in (0,1)$, then
$$ \lVert \lambda v + (1-\lambda)w \rVert = \lambda \lVert v \rVert + (1-\lambda)\lVert w \rVert \tag{1}\label{1} $$
implies $v = w$, what happens if the norm is not strictly convex? For example, if we conser de 1-norm in $\mathbb{R}^2$
$$ \lVert (x,y) \rVert_{1} = |x| + |y| $$
then equality holds in \eqref{1} if and only if $v$ and $w$ are in the same quadrant (if you think in $\mathbb{R}^n$ it would be in the same hyperoctant), is there some kind of generalization of this for any norm? If there is and I would have to guess, I would say they have to be in the same polyhedral cone (and the shape of the cone is defined by the norm), but I can't find a way to prove this. I am thinking in norms in $\mathbb{R}^n$.