Projecting the unit cube onto a subspace [closed]

I have some (rather exotic) subspace $L<R^n$, and I want to show that every non-zero vector in $\{0,1\}^n$ has a relatively small projection onto $L$. What general results and tools can be helpful? Anything from geometry of numbers?

Any suggestions appreciated!

Upon looking at the responses, some explanations may be in order. Let's say that a subspace $L<R^n$ is oblique if for any vector $z\in\{0,1\}^n$, the projection of $z$ onto $L$ is of length at most $\|z\|/\log\log n$ (say). What properties of a subspace can ensure that it is oblique? Can any general "obliqueness criteria" be given?

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Please state a reasonable question and give background. As is, I vote to close. – Bill Johnson Apr 4 2011 at 20:39
I second Bill's vote. I don't care much about the background but a legitimate question should be far more precise than this. As of now, I do not even understand what "relatively small" might mean here. Indeed, no vector is longer than $\sqrt n$ and, unless your subspace is more or less aligned with some coordinate plane of small dimension, there is little chance to get much less for the maximal projection. – fedja Apr 5 2011 at 3:14
@Bill: I believe, it is reasonable to request background for extremely specialized and artificially looking questions. This is certainly not the case with a general question like this. The question is motivated by my attempt to construct a graph with some particular property, but I am certainly not in a position to describe here all the work which led me to this question. – Seva Apr 5 2011 at 6:50
@fedja: I explained above what I mean by "relatively small". As to you last remark: if the subspace is aligned with some coordinate plane, then it most certainly does not have the property in question. Loosely speaking, what I need is a criterion to show that a subspace is not aligned with the coordinate planes! – Seva Apr 5 2011 at 6:53
I suggest re-editing or posting a new question with a description of the invariant subspace L and the clarification that you want the ratio. – Aaron Meyerowitz Apr 5 2011 at 13:12

closed as not a real question by Bill Johnson, Igor Rivin, fedja, Andreas Thom, Daniel LittApr 5 2011 at 4:49

It would help to know what $L$ is.

update This answer was for the question of how short the longest projection could be if $L$ is allowed to vary. There have since been some clarifications.

Here is an upper bound which I think I can prove to be the minimum among all the $d$ dimensional spaces of $\mathbb{R}^n$ (with $n>d$ of course)

Let $L$ be the subspace of $\mathbb{R}^n$ consisting of vectors whose first $d+1$ coordinates sum to $0$ and whose remaining coordinates are $0$. Project the vectors of $\lbrace0,1\rbrace^n$ onto $L.$ The length of the longest projection is $$\begin{cases} \sqrt{\frac{d+1}{4}}, & \mbox{if }d=2k+1 \\ \\ \sqrt{\frac{d+1-\frac{1}{d+1}}{4}}, & \mbox{if }d=2k. \end{cases}$$

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Thanks for an attempt to answer in essence - but I am afraid, I don't understood much here. For $d=1$ you seem to claim that the longest projection of a vector from $\{0,1\}^n$ onto your subspace is $0$, do you? Anyway, I am speaking about some particular subspace, and I need the length of the projection normalized by dividing by the length of the vector itself. – Seva Apr 5 2011 at 7:02
Concerning "what $L$ actually is": is is an invariant subspace of a (high) tensor power of the "Fibonacci matrix" $\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}$. I am not asking, however, to look at my particular problem; rather, I am to trying to figure out what are possible approaches and useful tools. – Seva Apr 5 2011 at 7:04
OK, thanks, I think I fixed it. For $d=1$ the subspace spanned by $[1,-1,0,0,0,\cdots]$ sends some things to the zero vector and others to $[1/2,-1/2,0,0,0,\cdots]$. For $d=2$ some things go to $[2/3,-1/3,-1/3,0,0,0,\cdots]$. For $d=3$ some things go to $[1/2,1/2,-1/2,-1/2,0,0,\cdots]$. – Aaron Meyerowitz Apr 5 2011 at 8:31
I see your point - but my question is not about the subspaces that you consider! I don't want all projections to be (relatively) small for any vector $z\in\{0,1\}^n$ and any subspace $L$ - I have some particular subspace $L$ in mind, and I wonder what properties of this my subspace can guarantee that it is "not aligned with the vectors from $\{0,1\}^n$". – Seva Apr 5 2011 at 9:03
I am afraid there is still some misunderstanding here. Let's say that a subspace $L<R^n$ is good if for any $z\in\{0,1\}^n$, the projection of $z$ onto $L$ has length at most $\|z\|/\log\log n$. Good subspaces do exist: say, the one-dimensional subspace spanned by the vector $(1,1/\sqrt2,...,1/\sqrt n)$ is good. What I seek is a sufficiently general and versatile criterion for a subspace to be good. – Seva Apr 5 2011 at 14:10