Blaschke-type conditions in unbounded domains, generalized convexity and applications in perturbation theory

We introduce a new geometric characteristic of compact sets on the plane called $r$-convexity, which fits nicely into the concept of generalized convexity and extends essentially the conventional convexity. For a class of subharmonic functions on unbounded domains with $r$-convex compact complement, with the growth governed by the distance to the boundary, we obtain the Blaschke–type condition for their Riesz measures. The result is applied to the study of the convergence of the discrete spectrum for the Schatten–von Neumann perturbations of bounded linear operators in the Hilbert space.