A note on the boundary of the Birkhoff-James ε-orthogonality sets

Abstract

The Birkhoff-James $\varepsilon$-sets of vectors and vector-valued polynomials (in one complex variable) have recently been introduced as natural generalizations of the standard numerical range of (square) matrices or operators and matrix or operator polynomials, respectively. Corners on the boundary curves of these sets are of particular interest, not least because of their importance in visualizing these sets. In this paper, we provide a characterization for the corners of the Birkhoff-James $\varepsilon$-sets of vectors and vector-valued polynomials, completing and expanding upon previous exploration of the geometric propertiesof these sets. We also propose a randomized algorithm for approximating their boundaries.