Inscribed and circumscribed radius of \(\kappa \)-convex hypersurfaces in Hadamard manifolds (Q6642735)

The work entitled ``Inscribed and circumscribed radius of \(\kappa\)-convex hypersurfaces in Hadamard manifolds'', represents a significant extension of Blaschke's theorems in negative curvature spaces, generalizing classical results and building a theoretical framework useful for future developments. The main proofs are well structured and rely on consolidated techniques of Riemannian and hyperbolic geometry. The results offer useful tools for the study of convex-smooth and polygonal surfaces in Hadamard manifolds, an active and relevant field in differential geometry.\N\NHowever, I found some points that need revision:\N\N1) In \textit{Theorem 1.7}, the condition \(k_2 \coth(k_2 \rho)\geq k_1\) is crucial for the result, but a detailed explanation of why this inequality emerges naturally from the geometric assumptions is missing. Although the connection with the comparison of curvatures is intuitive, a brief discussion on the geometric meaning of this condition would strengthen the reader's understanding.\N\N2) The explicit dependence of the upper bounds on \(\rho\) might appear counterintuitive to some experts. I would suggest adding a remark on the role of \(\rho\) as a parameter that mediates between curvature and geometric dimension of the polygon or surface.\N\N3) The use of \textit{Lemma 2.1} (from reference [5]) to obtain the main result is correct, but the transition from the assumptions of the lemma to the specific case of the paper is only implicitly justified. A brief clarification on how the \(k_2\)-convexity conditions exactly satisfy the requirements of the lemma could clear up any doubts.\N\N4) In \textit{Remark 4.4}, the authors acknowledge that the upper bound for \(r\) is not optimal and that a better result should be for \(R\). It would be helpful to explicitly highlight how to improve these estimates or to discuss why improvement is not possible in this context.\N\N5) In \textit{Remark 4.2}, the authors treat the case \(k_2 = 0\) by modifying the definition of curvature \(\kappa_A\), but do not elaborate on the implications of this choice on the geometric properties of the problem. Although this point does not affect the main results, I would suggest clarifying how the modified definition of \(\kappa_A\) interacts with the general assumptions and whether there are limits to its applicability.\N\N\textbf{Conclusion:}\N\NThe paper is well written and mathematically rigorous. Clarifying the aspects listed above could further strengthen the robustness of the work. Overall, the paper is worthy of publication after minor revisions.