Compact pseudoconcave sets

TL;DR

This paper investigates properties of compact pseudoconcave sets, demonstrating their core can be smaller than the set itself, and exploring their decomposition and behavior of plurisubharmonic functions.

Contribution

It answers three open questions about compact pseudoconcave sets, revealing new structural and functional properties.

Findings

The core of a compact pseudoconcave set can be strictly smaller than the set.

The core of a compact set must be pseudoconcave.

Such sets can be decomposed into pseudoconcave components where all smooth plurisubharmonic functions are constant.

Abstract

Replying to three questions posed by N. Shcherbina, we show that a compact psudoconcave set can have the core smaller than itself, that the core of a compact set must be pseudoconcave, and that it can be decomposed into compact pseudoconcave sets on which all smooth plurisubharmonic functions are constant.