Elements of Minimal Norm on an affine subset of a Banach Space 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.
 A: Since $||v_1+tv_2||$ is a convex function of $t$, it is differentiable almosst everywhere and attains its minimum at the points where its derivative changes sign or is zero (the minimum does not have to be unique). If your data provides enough information to find such point, you are done. In general, you need to have a sufficient amount of data to say something about the problem.
A: the most important fact to remember is probably that 
for a linear subspace $S$ of a Banach space $X$, the projection operator : $$P_S(x) = \inf_{u \in S} \|x-u\|$$
is sublinear :  (the result of the minimization is $u$, but for convenience assume the result can also be $\|x-u\|$ )
$$P_S(\alpha x) = \inf_{\alpha u \in S} \|\alpha x-\alpha u\| = |\alpha|\inf_{\alpha u \in S} \| x-u\| = |\alpha| P_S(x)$$
$$P_S(x+y) = \inf_{u,v \in S} \|x+y-(u+v)\| \le \inf_{u,v \in S} \|x-u\| + \|y-v\|$$ $$= \inf_{u \in S} \|x-u\|+\inf_{v \in S} \|y-v\| = P_S(x)+P_S(y)$$
I'm pretty sure this is why we require in the Hahn–Banach theorem that some linear map on a linear subspace $S$ of $X$ be bounded (on $S$) by some sublinear function on $X$.
