Homeomorphisms of Compact, Convex Sets and the Jacobian Matrix

Let K Rⁿ be a compact, convex polyhedron and f: K Rⁿ a C¹ 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 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.