Product of diagonal entries of the unitary orbit of a 3-by-3 normal matrix

Let N be a 3×3 normal matrix. We investigate the sets where U(3) is the group of 3×3 unitary matrices and 1⩽k⩽3. Geometric properties of these sets are studied, namely, star-shapedness and simple connectedness are investigated. A method for the numerical estimation of is also provided for normal matrices of size 3.