# Minimizing sum of square lengths of vectors under Projections onto Planes in R^n

Let $e_i=(0,...,0,1,0,...0)$, be the standard unit basis vectors in $R^n$. Consider the set of $n\choose 2$ vectors $v_{i,j}=e_i-e_j$ where $i < j$. For any fixed $0\leq k\leq {n\choose 2}$. Let $P$ be a projection onto a plane, which is perpendicular to $(1,....1)$ and let $|A|=k$, where $A\subset \{\{1,...n\}^2 : i<j \}$.

I am interested obtaining upper bounds, in terms of $k$ and $n$, on $max_{P}\{\sum _{(i,j)\in A} \|P(v_{i,j}) \|^2\}$.

I have tried using Mathematica to get some idea of the bound, but one problem was parametrizing the planes.

There is an obvious bound of $2n$, which actually corresponds to $k=n$, since increasing $k$ only increases the bound.

Any references or suggestions would be very helpful.

Thanks very much!

-