Star-convexity of the eigenvalue regions for stochastic matrices and some subclasses

Star-convexity of the eigenvalue region for the set of n × n stochastic matrices has already been proved, for n ≥ 2, by Dmitriev and Dynkin. The star-convexity property enables full determination of the eigenvalue region by its boundary.
This study offers a more straightforward proof that extends to other subclasses of the stochastic matrices. Furthermore, the proof is constructive as it includes the explicit construction of the corresponding realizing matrices. Explicit sufficient conditions for star-convexity of the eigenvalue regions of stochastic subclasses are presented. In particular, star-convexity of the eigenvalue region is proved for the n × n doubly stochastic and the n × n monotone stochastic matrices.