Say that a set $X \subset \mathbf{S}^{d-1}$ is *ortho-closed* if for any set $\{x_1,\dots,x_{d-1}\} \subset X$ there exists $x \in X$ such that $\langle x,x_i \rangle = 0$ for $i=1,\dots,d-1$. (Apologies for the terrible name - other suggestions are welcome.) Thus, as a silly example, the set of standard basis vectors is ortho-closed.

My rather vague question is, what (non-trivial) finite ortho-closed sets are there? Much more concretely (and actually what I'm most interested in), with $d=3$, is there a finite ortho-closed set containing the normalizations of the seven non-zero 0-1 vectors in $\mathbb{Z}^3$? (Or indeed is there a finite ortho-closed set containing an arbitrary fixed set?)

It seems to me that the answer ought to be `no', and that one ought to be able to construct an infinite sequence of vectors that would have to be in any such set. However I can't yet see how to do this, and it occurred to me that this is a fairly natural property which might already be known. Thus, any references would also be greatly appreciated.