Unions of \(n\)-planes as polynomial envelope with nonempty interior
The union of two totally real \(n\)-dimensional vector spaces (or \(n\)-planes) of \(\mathbb{C}^n\), which contain 0, is not always polynomially convex for \(n \geq 2\). \textit{P. Thomas} [Proc. Symp. Pure Math. 52, Part 1, 233-244 (1991; Zbl 0755.32011)] gave an estimate of the minimal number of planes containing 0, and such that the hull of their union is \(\mathbb{C}^n\). This estimate has an exponential growth with \(n\). We show here the following result: Theorem. For every \(n \geq 2\), there are \(n + 1\) planes \(P_1,\dots, P_{n + 1}\) in \(\mathbb{C}^n\) containing 0 and such that the hull of \(\bigcup^{n+1}_{j = 1} \;P_j\) has a nonempty interior.
scientific article; zbMATH DE number 856580