2207.01901
ON VARIATIONAL PRINCIPLE FOR UPPER METRIC MEAN DIMENSION WITH POTENTIAL
Rui Yang, Ercai Chen, Xiaoyao Zhou
correctmedium confidence
- Category
- Not specified
- Journal tier
- Specialist/Solid
- Processed
- Sep 28, 2025, 12:56 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper proves a variational principle for upper metric mean dimension with potential by a clean Hahn–Banach separation and Riesz representation argument and defines F(µ) via A = {g : mdim_M(T, −g, d)=0}, yielding mdim_M(T,f,d) = sup_{µ∈M(X)} [F(µ)+∫ f dµ] (Theorem 1.1 and Definition 3.4). The candidate solution follows a different convex-analytic route at fixed scales ε and gets the ≥ inequality correctly, but its derivation of the ≤ inequality relies on the false interchange limsup_ε sup_µ ≤ sup_µ limsup_ε, which is generally reversed. Hence the model’s proof is incomplete/incorrect, while the paper’s argument is logically complete and consistent with its stated hypotheses.
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions
\textbf{Journal Tier:} specialist/solid
\textbf{Justification:}
The paper delivers a rigorous and well-structured variational principle for upper metric mean dimension with potential using classical convex-analytic tools (separation and Riesz representation). The presentation is generally clear and the result is valuable for the field. Minor edits could further improve readability and contextualization, especially contrasting the separation method with a potential Fenchel–Moreau viewpoint.