Breakthrough Tracker record
A Lean-certified counterexample answers Pommerenke's convexity question negatively
Yuankai Guo and Xiaozhe Hu construct explicit convex functions in the exterior-disk class Σ whose convex linear combination remains univalent but is not convex, answering Pommerenke's 1962 question in the negative.
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- Stable ID
math-pommerenke-convex-combination-counterexample-2026- Revision
math-pommerenke-convex-combination-counterexample-2026.v1- Field
- Mathematics · Complex analysis, univalent functions and formal verification
- Evidence
- Tier 1 · Peer reviewed: No
- Record state
- Provisional · Provisional formally certified counterexample
- Last checked
AI role
No substantive AI role was disclosed in the inspected manuscript or repository description.
Record details
- Problem or result
- Pommerenke's Problem 6.10 on convex combinations in the exterior-disk class Σ
- Authors
- Yuankai Guo and Xiaozhe Hu
- Institutions
- No current institutional affiliations stated in the manuscript
- Result date
- Preprint submitted September 3, 2026
Why it matters
The explicit algebraic construction closes a question carried in Hayman's problem collection and pairs a short human-readable argument with a substantial formal proof artifact.
Limits
This is a six-page, non-peer-reviewed first version. The authors report 32 Lean files with no custom axioms or unfinished placeholders, but this maintenance cycle did not independently rebuild the repository.
Sources
- Primary: Guo and Hu, Pommerenke counterexample
- Artifacts: Lean 4 formalization of the Pommerenke counterexample
Correction and revision history
- 2026-09-07 — Added after primary-source, scope, status, AI-role and limitation review.
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