# Construction of labyrinths in pseudoconvex domains

## TL;DR

The paper constructs special convex sets within pseudoconvex domains in complex space, ensuring certain paths approaching the boundary have infinite length, thus solving an open problem in complex analysis.

## Contribution

It introduces a method to build convex sets in pseudoconvex domains that influence the behavior of boundary-approaching paths, addressing a previously unresolved problem.

## Key findings

- Constructed convex sets in pseudoconvex domains with specific boundary properties
- Ensured paths approaching the boundary while avoiding these sets have infinite length
- Solved an open problem from a recent research paper

## Abstract

We build in a given pseudoconvex (Runge) domain $D$ of $\mathbb{C}^N$ a $\mathcal O(D)$ convex set $\Gamma$, every connected component of which is a holomorphically contractible (convex) compact set, enjoying the property that any continuous path $\gamma:[0,1)\rightarrow D$ with $\lim _{r\rightarrow 1}\gamma(r)\in \partial D$ and omitting $\Gamma$ has infinite length. This solves a problem left open in a recent paper by Alarc\'on and Forstneri\v{c}.