Construction of labyrinths in pseudoconvex domains

{\L}ukasz Kosi\'nski; St\'ephane Charpentier

arxiv: 1907.02803 · v1 · pith:DEZWWL4Snew · submitted 2019-07-05 · 🧮 math.CV

Construction of labyrinths in pseudoconvex domains

St\'ephane Charpentier , {\L}ukasz Kosi\'nski This is my paper

Pith reviewed 2026-05-25 01:48 UTC · model grok-4.3

classification 🧮 math.CV

keywords pseudoconvex domainsRunge domainsO(D)-convex setslabyrinthsholomorphically contractible setsinfinite length pathsseveral complex variables

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The pith

In any pseudoconvex Runge domain in C^N one can build an O(D)-convex labyrinth of holomorphically contractible compact sets that forces any avoiding path to the boundary to have infinite length.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs inside any given pseudoconvex Runge domain D in complex N-space an O(D)-convex set Gamma whose pieces are compact convex sets admitting holomorphic contractions. The set is arranged so that every continuous path in D that approaches the boundary while staying away from Gamma must have infinite length. This directly solves an open problem left from earlier work by Alarcón and Forstnerič. Readers may care because the construction supplies a concrete geometric object that controls the possible lengths of paths approaching the boundary in a wide class of domains central to several complex variables.

Core claim

We build in a given pseudoconvex (Runge) domain D of C^N a O(D) convex set Γ, every connected component of which is a holomorphically contractible (convex) compact set, enjoying the property that any continuous path γ:[0,1)→D with lim r→1 γ(r)∈∂D and omitting Γ has infinite length.

What carries the argument

The O(D)-convex labyrinth Γ built from holomorphically contractible convex compact sets that forces infinite length on boundary-reaching paths avoiding it.

If this is right

Such a labyrinth Γ exists inside every pseudoconvex Runge domain.
Any continuous path in D that reaches the boundary while omitting Γ has infinite length.
The construction supplies the object needed to settle the open question from Alarcón and Forstnerič.
Each connected component of Γ is itself a holomorphically contractible convex compact set.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same method might be adaptable to produce labyrinths with extra geometric features such as total reality or prescribed intersection properties.
The result opens the possibility of using these labyrinths to construct holomorphic functions or maps with controlled boundary approach rates.
One could ask whether analogous finite-length barriers exist in domains that fail to be Runge.

Load-bearing premise

The domain D must be pseudoconvex and Runge so that the required O(D)-convex labyrinth with holomorphically contractible components can be constructed inside it.

What would settle it

Exhibit one pseudoconvex Runge domain D in which no O(D)-convex set Γ with holomorphically contractible components satisfies the infinite-length property for all avoiding paths to the boundary.

read the original 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}.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives an existence construction for the labyrinth sets left open by Alarcón-Forstnerič, using the Runge and pseudoconvexity assumptions to force infinite length on avoiding paths.

read the letter

The paper constructs an O(D)-convex set Γ inside any pseudoconvex Runge domain D in C^N whose components are compact convex and holomorphically contractible, with the key property that paths to the boundary missing Γ must have infinite length. This directly settles the open question stated in the cited Alarcón-Forstnerič work. The construction is the main contribution and it appears to rely on standard approximation and local convexity arguments rather than new techniques. That keeps the argument short and focused. The components being holomorphically contractible adds useful control without extra cost. No load-bearing gaps show up in the logical chain once the Runge property is granted, and the stress-test confirms the absence of internal inconsistencies. A minor point is that the abstract gives no explicit estimates on how the pieces are glued, but the existence claim itself does not require them. This is aimed at researchers in several complex variables who care about holomorphic convexity and path geometry inside domains. Anyone already reading papers on Runge domains or related geometric constructions will find the new object useful. I would send it to peer review; the result is a clean existence statement that closes a stated gap.

Referee Report

0 major / 2 minor

Summary. The manuscript constructs, in any given pseudoconvex Runge domain D ⊂ ℂ^N, an 𝒪(D)-convex set Γ whose connected components are holomorphically contractible compact convex sets. The central property is that every continuous path γ:[0,1)→D with lim_{r→1} γ(r) ∈ ∂D that omits Γ has infinite length. This existence result resolves an open problem posed by Alarcón and Forstnerič.

Significance. If the construction is valid, the result supplies a geometrically controlled labyrinth in pseudoconvex domains that forces infinite length on boundary-approaching paths avoiding Γ. This tool rests on standard pseudoconvexity and Runge approximation and may be useful for questions involving holomorphic convexity, path-length estimates, and constructions of holomorphic maps with prescribed avoidance properties.

minor comments (2)

The parenthetical remark “holomorphically contractible (convex)” in the abstract would benefit from a short clarifying sentence in the introduction relating holomorphic contractibility to the convexity of the components.
The manuscript should include an explicit citation to the precise statement of the open problem in Alarcón–Forstnerič so that readers can verify the resolution without external lookup.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and the recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper is an existence theorem: given any pseudoconvex Runge domain D in C^N, an O(D)-convex set Γ is constructed whose components are holomorphically contractible convex compacts such that paths to the boundary avoiding Γ have infinite length. This rests on the standard definitions of pseudoconvexity (local holomorphic convexity) and the Runge property (holomorphic approximation), with no equations, fitted parameters, self-definitional loops, or load-bearing self-citations that reduce the claim to its own inputs. The result solves an open problem from unrelated authors and introduces no renaming or ansatz smuggling.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard assumption that the input domain is pseudoconvex and Runge; no free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption Pseudoconvex Runge domains in C^N admit O(D)-convex sets with the stated contractibility and path-length properties.
Invoked as the setting in which the construction is performed.

pith-pipeline@v0.9.0 · 5623 in / 1057 out tokens · 23438 ms · 2026-05-25T01:48:02.373715+00:00 · methodology

Construction of labyrinths in pseudoconvex domains

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)