For vectors $x, y \in \mathbb{R}^n\setminus\{0\}$, under the Euclidean norm we have that $$\left\Vert x + y \right \Vert_2 = \left \Vert x \right \Vert_2 + \left \Vert y \right \Vert_2 \iff \exists c \in \mathbb{R}_{> 0} \textrm{ s.t } \ x = cy$$ $$$$
However, I've come to a point in my research where I can do some really cool things on a normed space where two vectors being on the same ray is not a requirment to satisfy equality in the triangle inequality. For example, under the $L^1$ norm, i.e $$\left \Vert x \right \Vert_1 = \sum_{i=1}^n \left \vert x_i \right \vert,$$ as in this answer we have something like \begin{gather}\left\Vert x + y \right \Vert_1 = \left \Vert x \right \Vert_1 + \left \Vert y \right \Vert_1 \\ \iff \\ \textrm{sgn}(x_{i}) = \textrm{sgn}(y_i) \ \textrm{for} \ i \ \textrm{s.t.} \ x_{i}, y_i \neq 0. \end{gather} $$$$ Additionally, under the $L^\infty$ norm, $$ \left \Vert x \right \Vert_\infty = \sup_{1 \leq i \leq n} \left \vert x_i \right \vert, $$ we have \begin{gather}\left\Vert x + y \right \Vert_\infty = \left \Vert x \right \Vert_\infty + \left \Vert y \right \Vert_\infty \\ \iff \\ e_{k}^T x = \sup_n \vert x \vert \ \textrm{and} \ e_k^T y = \sup_n \vert y \vert, \ \textrm{for some standard basis vector } e_k \in \mathbb{R}^n,\end{gather}
Essentially, my question boils down to this: What are some other lesser known examples of normed spaces (besides the above two - they're the only examples I've been able to find) where equality holds in the triangle inequality for conditions other than two vectors being on the same ray? Any and all kinds of examples would be great, not just those concerning $\mathbb{R}^n$ and $L^p$ norms.
