On the Steinness index

TL;DR

This paper introduces the Steinness index, explores its equivalence to strong Stein neighborhood bases, relates it to the Diederich-Forn{ {a}}ss index on worm domains, and characterizes it for certain pseudoconvex domains.

Contribution

It establishes the equivalence between the Steinness index and strong Stein neighborhood bases, and provides explicit formulas and characterizations for specific domain classes.

Findings

Steinness index is equivalent to strong Stein neighborhood basis.

Explicit relation between Steinness and Diederich-Forn{ {a}}ss indices on worm domains.

Steinness index equals 1 for certain pseudoconvex domains with boundary points of infinite type.

Abstract

We introduce the concept of Steinness index related to the Stein neighborhood basis. We then show several results: (1) The existence of Steinness index is equivalent to that of strong Stein neighborhood basis. (2) On the Diederich-Forn{\ae}ss worm domains in particular, we present an explicit formula relating the Steinness index to the well-known Diederich-Forn{\ae}ss index. (3) The Steinness index is 1 if a smoothly bounded pseudoconvex domain admits finitely many boundary points of infinite type.