Domains of unicity (Q1996449)

If \(\mathbf{F}=(F_{i}(x_{1},\dots,x_{n}))^{n}_{i=1}\) is a differentiable mapping from a subset \(K\) of \(\mathbb{R}^{n}\) into \(\mathbb{R}^{n}\) and if the Jacobian \((\partial F_{i}/\partial x_{j})\) of \(\mathbf{F}\) does not vanish at a point, then \(\mathbf{F}\) is univalent in a neighborhood of that point. This paper deals with the global univalence of \(\mathbf{F}\). \textit{D. Gale} and \textit{H. Nikaidô} proved in 1965 [Math. Ann. 159, 81--93 (1965; Zbl 0158.04903)] that if the Jacobian of \(\mathbf{F}\) is a \(P\)-matrix at every point of \(K\) and \(K\) is a closed rectangular region, then \(F\) is injective on \(K\). Under the stronger hypothesis that the Jacobian is positive definite on \(K\), the same conclusion is valid on any convex set \(K\). Here, the author addresses the question on what domains are the above two results true. One shows that the rectangular domains and the convex sets are the only ones for which the Gale-Nikaido results are true. Recall that a square matrix is called a \(P\)-matrix if all its principal submatrices have a positive determinant. Also we remember that a square matrix \(A\) is called positive definite if \(\mathbf{x}^{*}A\mathbf{x}>0\) for all non-zero vectors \(\mathbf{x}\), where \(\mathbf{x}^{*}\) is the transposed of \(\mathbf{x}\). Then one gets the following results: \begin{itemize} \item Let \(K\subset \mathbb{R}^{n}\) be a non-empty compact set with the property that any \(C^{1}\) mapping \(\mathbf{F}: K\to \mathbb{R}^{n}\), for which the Jacobian is a \(P\)-matrix at every point of \(K\), is univalent on \(K\). Then \(K\) is a closed rectangular region. \item Let \(K\subset \mathbb{R}^{n}\) be a non-empty compact set with the property that any \(C^{1}\) mapping \(\mathbf{F}: K\to \mathbb{R}^{n}\), for which the Jacobian is positive definite at every point of \(K\), is univalent on \(K\). Then \(K\) is convex. \end{itemize}