Condition for the numerical range to contain an elliptic disc
For an \(n\times n\) matrix \(A\) and an elliptic disc \(E\) in the plane, it is shown that the sum of the number of common supporting lines and the number of common intersection points of the supporting lines with \(E\) and the numerical range \(W(A)\) should be at least \(2n+1\) in order that \(E\) be contained in \(W(A)\). This generalizes previous results of \textit{Joel Anderson} [unpublished, cf. \textit{M. Radjabalipour} and \textit{H. Radjavi}, Pac. J. Math. 61, 507-511 (1975; Zbl 0323.47004)] and \textit{R. C. Thompson} [ibid. 254, 19-28 (1997; Zbl 0880.15032)]. As an application, this result is used to verify a special case of the Poncelet property conjecture that if \(E\) is an elliptic disc contained in the unit open disc \(\mathbb{D}\), then \(E\) is the numerical range of some matrix \(A\) in the class \(S_n\) if and only if it has the property that for any point \(\lambda\) in \(\partial \mathbb{D}\) there is an \((n+1)\)-gon interscribing between \(\partial\mathbb{D}\) and \(\partial E\) and having \(\lambda\) as a vertex. As noted in ``Added in proof'', the Poncelet property conjecture itself (without assuming \(E\) is an elliptic disc) is refuted by \textit{B. Mirman} [Linear Algebra Appl., 360, 123-150 (2003; Zbl 1028.15025)]. In the Appendix, a proof of the main result based on Bézout's theorem is given.
scientific article; zbMATH DE number 1903586