A geometric Hall-type theorem

We introduce a geometric generalization of Halls marriage theorem. For any family $F = {X_1, dots, X_m}$ of finite sets in $mathbb{R}^d$, we give conditions under which it is possible to choose a point $x_iin X_i$ for every $1leq i leq m$ in such a way that the points ${x_1,...,x_m}subset mathbb{R}^d$ are in general position. We give two proofs, one elementary proof requiring slightly stronger conditions, and one proof using topological techniques in the spirit of Aharoni and Haxells celebrated generalization of Halls theorem.