Can anybody help me with this? Is the following statement true? Can anybody prove it (or prove it wrong)?
For $m<n$ let $\mathcal{M}:=\{A=(v_1,\ldots,v_m)\in \mathbb{R}^{n\times m}:\; <v_i,v_j>=\delta_{ij} \;\forall i,j\}$.
Moreover, let $\mathcal{A}\subset \mathcal{M}$ be a set with the property $\det(A^tB)>0 \; \forall A,B\in \mathcal{A}$.
Then there exists a $\theta=\theta(n)>0$ and a $C\in \mathcal{M}$ such that $\det(A^tC)>\theta \; \forall A\in \mathcal{A}$.
Thanks in advance for any help!