Let $K \subset R^n $ be a compact, convex polyhedron and $f:K \to R^n $ a $C^1 $ function. The problem of existence of a global inverse for f is studied. It is shown (Theorem 1) that f has an inverse, if, for every $x \in K$, the Jacobian of f at x, $Jf(x)$, is such that for every linear space spanned by a face of K containing x the determinant of the linear map from L to L formed by projecting $Jf(x)$ on L has positive sign. Theorem 2 is a similar result for K with smooth boundary. The theorems generalize the well-known Gale–Nikaido theorems, which originated in some problems of mathematical economics.