  We prove a geometric property of the set A^{-1} of inverses of the nonzero
elements of an F_q-subspace A of a finite field involving the size of its
intersection with two-dimensional F_q-subspaces. We give some applications,
including a new upper bound on the size of the intersection of A^{-1} and B
when A and B are F_q-subspaces of different dimension of a finite field,
satisfying a suitable natural assumption.
