Controlled inflation of smooth Jordan domains #
A bounded smooth convex Jordan domain can be enlarged by an arbitrarily small homothety about an interior point. Convexity puts the old closed carrier strictly inside the enlarged open carrier, while boundedness controls the Hausdorff displacement. Thus closed smooth Jordan carriers themselves have the local outer-approximation property used by the terminal exhaustion.
The closed carrier of a smooth Jordan domain admits arbitrarily tight smooth Jordan outer approximations.
theorem
crouzeix_palencia_of_closure_numericalRange_eq_smoothJordanClosure
{E : Type u}
[NormedAddCommGroup E]
[InnerProductSpace ℂ E]
[CompleteSpace E]
(A : E →L[ℂ] E)
(Omega : SmoothJordanDomain)
(hW : closure (numericalRange A) = closure Omega.carrier)
:
IsKPolynomialSpectralSet A (1 + √2) (closure (numericalRange A))
If the closed numerical range is already the closed carrier of a smooth convex Jordan domain, controlled inflation gives the exact Crouzeix--Palencia spectral-set bound.