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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.

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More precisely I mean that $X$ is a real banach space, and $t$ ranges over the reals. I would like to know if one could explicitly find a real valued continuous function of $v_1$ and $v_2$ of absolute value strictly less than $||v_1 +tv_2||$ for all $t$. – student Oct 16 '13 at 15:08
Of course $\ f(v_1\ v_2) := -1\ $. Seriously, it's perhaps my own problem but I give up on many questions--I wish they would be summarized by a formal logical statement like $\ \forall_{a\ b}\exists_{c}\ \ldots\ $ and similar. – Włodzimierz Holsztyński Oct 16 '13 at 15:31
I'm sorry for being imprecise, but I meant a non-negative and (non-trivial) function. More strongly, is there a formula for the infimum over all $t$ of $||v_1+tv_2||$? – student Oct 17 '13 at 1:28
Thank you, @edger. You mean to discuss a generalization of cosinus from Hilbert spaces onto Banach spaces. One stage should be concerned with the 2-dimensional Banach spaces only. The second stage would address arbitrary dimension. That's how I see it. – Włodzimierz Holsztyński Oct 17 '13 at 15:46
This q. is about cosinus, sinus and a norm. Given $\ p\ v\in S\subseteq B$ of elements of norm $1$ one wonders about $\ ||a||_v\ :=\ \inf_{t\in\mathbb R} \sqrt{t^2+||p-t\cdot v||^2}\ $, etc. – Włodzimierz Holsztyński Oct 17 '13 at 16:16

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