Two remarks on the Suita conjecture

TL;DR

This paper investigates the Suita conjecture, demonstrating its failure in certain multidimensional non-pseudoconvex domains while confirming a related converse in planar cases.

Contribution

It provides the first proof of the weak converse to the Suita conjecture for finitely connected planar domains and shows the conjecture's failure in specific multidimensional settings.

Findings

Weak multidimensional Suita conjecture fails for non-pseudoconvex domains.

Weak converse to the Suita conjecture holds for finitely connected planar domains.

Failure of the conjecture in certain smooth, non-pseudoconvex domains.

Abstract

It is shown that the weak multidimensional Suita conjecture fails for any bounded non-pseudoconvex domain with $C^{1+\varepsilon_-}$-smooth boundary. On the other hand, it is proved that the weak converse to the Suita conjecture holds for any finitely connected planar domain.