About the projection of a set of vectors

This problem is about the projection of a set of vectors onto another set of vectors. And tries to bound the maximum value of the minimum possible projection.

Suppose I have a set of k normalized orthogonal vectors, each being n-dimensional. And, suppose further that these vectors w_1,...,w_k lie on the principal axes. That is, w_i = (0,...,1,0,...,0), having 1 on the ith element and 0 otherwise, for all k >= i >= 1.

I have another set of normalized orthogonal m vectors V = v_1,...,v_m, where m <= k-1.

The problem is this: Can you put a bound on the maximum value of the minimum length projection of one of the w_i's onto the subspace spanned by v_i's? Specifically, I am asking an upper bound on

c = max_{j}^k min_{V} {projection of w_j onto V}

CAUTION: This, as I have figured out, is not a trivial problem. It may not even be elementary. I just want to know whether it has been proved before so that I don't waste my time. Any comments, suggestions and direction are much

appreciated.

Pigeonhole principle seems to play an important role. But, the set V can be arbitrary complicated which is hard to analyze.

My conjecture:

For i <= log(k), c <= i/k. (straight-forward pigeonhole)

For i > log(k), c <= log(i)/k. (which means that when i is big enough, we have a stronger bound than what we get from pigeonhole.)