Let $v_1$ and $v_2$ be two elements of a real Banach space $(X,\lVert\;\rVert)$, each of unit norm. Consider the one-dimensional affine subspace $v_1+tv_2$ for real $t$. Is there a general formula for the infimum $$\inf_{_t\in \mathbb{R}}||v_1+tv_2||$$ explicitly in terms of $v_1$ and $v_2$?

For example, in the case where $X$ is a Hilbert Space, it can be shown that the required formula is obtained when $t=-\langle\;v_1,v_2\;\rangle$.

I doubt there is such a formula. If the norm is differentiable, or can be made differentiable by composing it with some monotone differentiable function of $\mathbb{R}^{+}$, then perhaps there would be a chance at obtaining such a formula (simply by differentiating, setting that equal to zero, and trying to solve for the specific $t$).

If this is asking too much, it would also be nice to know if one can explicity write down a function $f(v_1,v_2)$ such that $$0 \leq f(v_1,v_2) \leq \inf_{_t\in \mathbb{R}}||v_1+tv_2||$$ and such that $f(v_1,v_2)=0$ only if $v_1$ and $v_2$ are parallel.