I have heard that there exists the following conjecture (if I am not mistaken).


Let $u\_1,\dots,u\_n$ be unit vectors in an $n$-dimensional Euclidean vector space. Then there exists another unit vector $x$ such that 
$$\sum\_{i=1}^n |( x,u\_i)|\geq \sqrt{n}.$$



> 
> I am looking for a reference for this conjecture. Also I will be happy to know what is known about it. 
> 
> 
> 



Reference to a conjecture on unit vectors in Euclidean space
That isn't a conjecture but a routine exercise assigned after the students learn about Bang's solution of the Tarski plank problem. The proof goes in 2 steps:


1) Consider all sums $\sum\_j \varepsilon\_i u\_i$ with $\varepsilon\_i=\pm 1$ and choose the longest one. Replacing some $u\_j$ with $-u\_j$ if necessary, we can assume WLOG that it is $y=\sum\_i u\_i$. Comparing $y$ with $y-2u\_i$ (a single sign flip) we get 
$$
\|y\|^2\ge \|y-2u\_i\|^2=\|y\|^2-4\langle y,u\_i\rangle+4\|u\_i\|^2
$$
whence $\langle y,u\_i\rangle\ge 1$ for all $i$. (That part is the main step in the solution of the plank problem). 


2) Now we have $\|y\|^2=\sum\_i\langle y,u\_i\rangle\ge n$, so for $x=\frac y{\|y\|}$, we get
$$
\sum\_i\langle x,u\_i\rangle=\sqrt{\sum\_i\langle y,u\_i\rangle}\ge \sqrt n
$$
The End :-)


