Page 20 - Textos de Matemática Vol. 44

10 A. KOVACˇEC
a supporting line of ∆iC(A); so it is either a σ point or there exists a curve contained in ∆iC (A) with zero curvature at zU . This last hypothesis is impos- sible because the boundary of the disk centered at w and of radius |zU − w| has nonzero curvature at zU . Thus zU is a σ-point. We have thus shown that ∆iC(A) is the intersection of all closed disks containing the σ points. It is in- tuitively evident (and proved in the literature) that the mentioned intersection is the convex hull of all the σ points. Hence ∆iC(A) ⊆ conv{zσ : σ ∈ Sn}, as wished.
Many years later this result was extended further [BKdP94]: if (A,C) is a pair of normal matrices so that A is positive definite and has distinct eigenvalues while the eigenvalues of C lie on a line through the origin, then ∆C(A) ∈ conv{zσ : σ ∈ Sn}. I take the opportunity to thank Nat´alia and Professor Jo˜ao de Providˆencia for having me allowed to collaborate on that article whose main ideas are due to them and, in fact, for having me introduced to the whole circle of ideas.
In a quite recent article on OMC for scalar multiples for unitary matrices in Linear Algebra Appl. 422, the indefatigable S. W. Drury writes: ‘The proof uses many ideas from [dPB86, BKdP94]’. Hardly can there be a higher praise than this.
4. One more look at Wc(A)
[BdP88, boundary of Wc(A) for normal A, nonreal c]: We mentioned in Section 2 the convexity result of Westwick concerning Wc(A) if c is real; if A is in addition normal, then
n
Wc(A) = conv{ αiγσ(i) : σ ∈ Sn}.
i=1
What can be said if A is normal but c is nonreal?
Call a point P of ∂Wc(A) 2-regular if there exists an open disk B with
center at P such that ΓB = B ∩ ∂Wc(A) is a simple arc having a bijective parametrization γ : R → C of class C2 satisfying γ′(t) ̸= 0 for all t. Call Wc(A) locally concave at P if there exists such a disk B so that for any two points P′,P′′ of ΓB the segment [P′,P′′] has no interior point of Wc(A).
The following is shown:
Theorem. Let A be normal, P ∈ ∂Wc(A) be a 2-regular point and assume A has no non-real conjugate pair of eigenvalues. Then either Wc(A) is locally at P a straight line segment or Wc(A) is at P locally concave.