I'm looking for discussion / literature recommendations of a (possibly slightly weird) measure function: perhaps a weighted version of taxicab distance? This is not really my area and so I'd be super grateful if someone can point me in the right direction.
Suppose I have a space $[0, 1]^n$, so vectors $\textbf{v}$ are just n-tuples with either 0 or 1 for each $v_i \in \textbf{v}$.
Then we define the distance $D$ between $\textbf{v}, \textbf{u}$ as follows:
$$D(\textbf{v}, \textbf{u}) = \sum_{i}^{n} d(v_i, u_i)$$
where crucially: $d(1, 0) \neq d(0, 1)$. Informally, the idea is going 'up' from zero to one can be more (or less) easy than going down from one to zero.
Is this a well-known distance measure? Any guidance would be really appreciated.
But this cannot be a true metric, because the symmetry condition is violated: $D(a,b) = D(b,a)\ \forall \{ a, b \}$. Also, the triangle requirement need not be obeyed: $D(a,b) + D(b,c) \geq D(a,c)$.