The nicely named “theorem of the alternative” of Paul Gordan states the following
“If is a linear subspace then either contains a vector with positive coordinates or the orthogonal complement contains a non-zero vector with non-negative coordinates.”
These options are mutually exclusive, hence the name of the theorem. Gordan gave a proof of this in 1873 which involved a very clever inductive argument. I just thought of a self contained very short proof of this theorem. In fact I will prove an equivalent statement
“If then either there is a vector such that the inner product is positive for all or the convex hull of contains the origin.”
The equivalence can be seen as follows. Let be the matrix with as its rows. Then the columns of span a subspace and this transforms the second statement into the first and vice versa.
Now let have minimal norm. If then contains the origin. If and is any point in the convex hull then the line segment from to lies in and contains no point with smaller norm than . So for all we have
This implies that and in particular this holds for all . Done!