{"id":"15ff9b49-338a-4fb5-b49b-45510802095f","arxiv_id":"2608.09606","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proposes Choquet extensions for set-pair functions and L^p-integrated global extension constants, proving a monotonicity inequality that unifies Cheeger, spectral, and isoperimetric bounds across graph limits, hypergraphs, manifolds, and metric measure spaces.","lead":"Choquet-type extensions turn set-based optimization and isoperimetric problems into function-based ones, and an L^p-integrated monotonicity inequality unifies many classical bounds across graph limits, hypergraphs, manifolds, and metric measure spaces. A generalist may read it because one abstract inequality generates many known and potentially new spectral and geometric estimates under one roof.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Example 5.1's identification of c_2(Φ_1,fJ·K_1) with graphon conductance h_W is unproved and rests on an unpublished kernel-reduction lemma; the graphon/graphing Cheeger corollaries depend on it.","rationale":"The abstract core appears sound: the proof of Theorem 4.4 (Section 6.3) correctly applies the Mazur map f↦f^t and Hölder with ps'=qt', and the admissible-family step is justified by odd homeomorphism of Bd. Theorem 4.12's lower bound is also coherent under |e^J|≤2 and concentration. The weakness lies at the interface with applications. The reader's weakest assumption is on target: concrete quantities are identified with the abstract constants without proof. I partially disagree with the phrasing: moving from J·K_p to fJ·K_p is a deliberate choice of a different denominator, not an invalid replacement; the load-bearing unproved claim is the equality c_2(Φ_1,fJ·K_1)=h_W and equation (10), cited to the author's [40] and [39]. Because the headline graphon Cheeger results use exactly that identification, the conditional verdict stands; no new reason to accept or reject beyond the reader's assessment.","tokens_in":34492,"tokens_out":32016,"duration_ms":277342,"concrete_test":"Independently compute, for the complete graphon W≡1 on [0,1] and for a two-block graphon W=α on [0,a]×[0,a] (with known h_W and λ_2), the three quantities: (i) c_2(Φ_1, J·K_1) with the intrinsic max norm; (ii) c_2(Φ_1, fJ·K_1) with the degree-weighted L^1 norm; (iii) inf_{nonconstant f} ∫W|f(x)-f(y)| / inf_t∫deg|f-t|. If (ii) and (iii) do not coincide, the kernel-reduction identification in Example 5.1 is false. If they do coincide, the paper should still state the exact lemma from [40] and verify its hypotheses for φ_xy, so the derivation is checkable.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The single most load-bearing step is the passage from the abstract constants of Theorems 4.4/4.12 to the concrete quantities in Section 5.1. In Example 5.1 the paper replaces the intrinsic norm |f|_{φxy}=max(|f(x)|,|f(y)|) by the degree-weighted p-mean norm fJ·K_p and then asserts c_2(Φ_1,fJ·K_1)=h_W and, more generally, equation (10), whose denominator is inf_t∫ deg(x)|f(x)-t|^p dx. These are not interchangeable definitions: c_2 is an inf-sup over genus-2 symmetric sets with an uncentered denominator, whereas h_W is a plain infimum with a centered denominator. The collapse of the genus-2 minimax to the centered Rayleigh quotient is a nontrivial theorem; the paper cites only the author's unpublished preprint [40] ('kernel reduction lemma') without stating or proving it. The same identification is needed to equate c_2(Φ_2,fJ·K_2)^2 with λ_2(L_W). If this fails, Corollaries 5.3 and 5.6 and the signed/hypergraph analogues do not follow from the abstract theory.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a Choquet-type extension theory for set-pair functions, introduces L^p-integrated Choquet extensions Phi_p and Psi_p with associated global extension constants c(Phi_p, J.K_q), and proves a monotonicity inequality (Theorem 4.4, and its sharpened version Theorem 4.12 under degree-related and concentration assumptions). The abstract framework is then applied to graphons, graphings, hypergraphs, Riemannian manifolds and metric measure spaces to derive Cheeger-type inequalities, spectral bounds for maxcut and bipartiteness ratio, bounds for hypergraph p-Laplacians, and inequalities involving torsional rigidity and p-Laplacian eigenvalues. The core abstract monotonicity inequality is proved through a Mazur-map argument combined with Holder's inequality, and Theorem 3.2 gives a functional reformulation of set-based fractional optimization via sub- and super-Choquet extensions.","tokens_in":2625,"tokens_out":3496,"duration_ms":103002,"significance":"If the advertised applications are fully justified, the paper would provide a genuinely unifying framework: one monotonicity inequality generating Cheeger, spectral-gap, torsional-rigidity and isoperimetric estimates across very different structures is an appealing and potentially influential contribution. The abstract development, especially the disjoint-pair Choquet extension, the characterization of bisubmodularity by convexity (Theorem 2.9), and the L^p-integration of Choquet integrals, is original and likely to be useful beyond the specific applications. The proofs of Theorems 2.9, 3.2 and the main inequality (8) are largely coherent and checkable from the text; the paper also gives explicit functional representations for maxcut, bipartiteness ratio, conductance, frustration and Dirichlet p-isoperimetric constants. However, several load-bearing identifications between the abstract constants and concrete graphon/manifold quantities are only asserted, with key steps deferred to the author's own unpublished preprints ([38], [39], [40]), and a substantial block of applications in Section 5.4 is stated without proofs.","major_comments":[{"comment":"The central identification c_2(Phi_1, fJK_1)=h_W, and similarly c_2(Phi_2, fJK_2)^2=lambda_2(L_W), is asserted in Example 5.1 with the justification \"due to Theorem 3.2 and Example 3.6, as well as the kernel reduction lemma in [40]\". This is not a consequence of the definitions alone: Theorem 3.2 gives the quotient representation of h_W as an infimum over functions, whereas c_2(Phi_p, fJK_p) is a genus-2 inf-sup over symmetric sets, and the equality of these two quantities is a nontrivial minimax theorem. Moreover, the text replaces the intrinsic norm |f|_{phi_xy}=max(|f(x)|,|f(y)|) by the degree-weighted p-mean norm fJK_p, which is not an equality of norms but a change of the ambient norm used in the definition of c(.,.). Since Corollaries 5.3 and 5.6 and the later spectral bounds depend directly on Eq. (10), this identification is load-bearing and must be proved in the present manuscript, or the kernel reduction lemma of [40] must be stated and proved in sufficient detail for the reader to verify the equality.","section":"Section 5.1, Example 5.1 and Eq. (10)"},{"comment":"The proof of the strengthened monotonicity inequality rests on a lengthy estimate in Proposition 6.7, involving quantities s_{e,i}, I_e and e_phi,f that are introduced in the proof but not fully formalized; several inequalities in the chain are asserted without sufficient justification, and the final passage to the degree-related norm is not fully explained. Furthermore, Proposition 6.12, which is needed for the p=q=1 strengthening in Theorem 4.12, is proved in a single sentence invoking \"a statement similar to Theorem 3.2\". The theorem also relies on an inequality from [38, Lemma A.1] that is not stated in the paper. Given that Theorem 4.12 is the engine behind the Cheeger inequalities of Section 5.1, the proof must be made fully verifiable or the auxiliary results must be stated with complete proofs.","section":"Sections 4.2 and 6.4, Theorem 4.12 and Propositions 6.7, 6.11, 6.12"},{"comment":"The text states \"We omit the detailed proofs of all the results in this section\" immediately before Corollaries 5.22, 5.23 and Proposition 5.24. These results include the torsional-rigidity bound T_p(M) lambda_p(M) <= vol(M)^{p-1} and the lower bounds involving the distance function, which are among the paper's advertised applications. Moreover, the preceding identifications T_p(M)=c_1(nabla, L^1)^{-p} and lambda_p(M)=c_1(nabla, L^p)^p are not derived in the text. Since these results are presented as theorems of the abstract theory rather than as conjectures, the proofs, or at least the precise reduction from the abstract constants to the manifold quantities, should be included.","section":"Section 5.4"},{"comment":"The displayed alternative formula for the graphon conductance, with numerator 2|W|_1|f|_inf - integral W(x,y)|f(x)+f(y)| dxdy, is claimed to be equal to integral W(x,y)|f(x)-f(y)| dxdy. This equality is false in general: for nonnegative f, the alternative numerator equals integral W(x,y)(2|f|_inf - f(x)-f(y)) dxdy, which is generally strictly larger than integral W(x,y)|f(x)-f(y)| dxdy, the latter being integral W(x,y)(2 max(f(x),f(y))-f(x)-f(y)) dxdy. If the formula is intended as an upper bound, or holds only under additional conditions, this must be stated and proved; as written, the displayed equality in Corollary 3.9 item 3 is an internal inconsistency.","section":"Example 3.6 and Corollary 3.9, item 3"},{"comment":"Several load-bearing ingredients are cited to the author's own unpublished preprints: the kernel reduction lemma of [40] in Example 5.1, the inequality used in Proposition 6.11 as [38, Lemma A.1], and the (p,q)-Sobolev constant terminology of [39]. For a journal submission, these results should either be restated in the paper or the dependence should be made explicit enough that a reader can verify the claims without access to the preprints. The paper's novelty claim is also harder to assess while the boundary between new results and results imported from these preprints is not drawn precisely.","section":"General (dependence on preprints [38], [39], [40])"}],"minor_comments":[{"comment":"The notation for the integrated norms is garbled in the typesetting: the symbols qfy_p and qy_p appear to be intended as L^q-norms of |f|_{phi_e} and |f|_{psi_e}; these should be typeset correctly and defined in words.","section":"Definitions 4.1 and 4.2"},{"comment":"The distinction between J.K_p and fJK_p is essential (Theorem 4.4 versus Theorem 4.12), but the notation is not introduced with a dedicated table and the difference is easy to miss; a short notational glossary would substantially improve readability.","section":"Notation throughout"},{"comment":"The definitions of psi_2 in Example 3.4 and the computation bpsi_2(f)=2|f|_infty should specify how pairs involving the empty set are treated in the layer-cake integral, to avoid ambiguity about the value at t=0.","section":"Section 3.2.3 and Example 3.4"},{"comment":"The phrase \"this is equivalent to directly using fJK_p instead of JK_p\" is imprecise: the p-mean ((|f(x)|^p+|f(y)|^p)/2)^{1/p} is not equal to |f|_{phi_xy}=max(|f(x)|,|f(y)|); it is a different norm, and the equivalence is exactly what needs proof.","section":"Section 5.1, Example 5.1"},{"comment":"The expression in Eq. (14) defines c(Phi_p, fJK_q) only after specifying that the numerator uses |f(x)-f(y)|^p with respect to the measure eta induced by W; as written it is easy to misread the exponent of the numerator as missing a factor 1/p.","section":"Section 5.2, Eq. (14)"}],"recommendation":"major_revision","confidential_remarks":"The paper is broad and ambitious, and the abstract core is genuinely interesting. However, the advertised applications are not yet established because several critical identifications between the abstract extension constants and concrete graphon/manifold quantities are deferred to the author's own preprints or are asserted without proof, and Section 5.4 omits proofs entirely. There is also a displayed equality in Corollary 3.9 that appears to be false as stated. These issues are fixable in principle, but they require substantial additions rather than cosmetic changes. The heavy reliance on [38], [39], [40] should be flagged to the editor as a concern about verifiability and about the precise novelty attribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe thing to know about arXiv:2608.09606 is that the core abstract theory is interesting and largely coherent, but the advertised applications do not yet follow from it without extra, mostly unpublished, ingredients. The paper is a framework paper: a disjoint-pair Choquet extension, an L^p integration of Choquet integrals, global extension constants, and a monotonicity inequality (Theorem 4.4) that says c(Φ_p,J·K_q) ≤ t·c(Φ_{ps},J·K_{qt}) under ps'=qt'. The proof of that inequality is a Mazur-map plus Hölder argument, and it is clean. Theorem 3.2's functionalization of set-based fractional optimization is also proved properly and gives nice equivalent formulations for maxcut, bipartiteness, conductance, frustration, and Dirichlet isoperimetric constants. If those were the whole paper, I would be fairly positive.\n\nThe problems are in the applications. Example 5.1 replaces the intrinsic norm |f|_{φxy}=max(|f(x)|,|f(y)|) with the degree-weighted p-mean norm (fJ·K_p), then asserts c_2(Φ_1,fJ·K_1)=h_W and c_2(Φ_2,fJ·K_2)^2=λ_2(L_W), citing the author's unpublished \"kernel reduction lemma\" [40]. That is a nontrivial collapse of a genus-2 minimax into a plain centered Rayleigh quotient; the paper does not state or prove it. The Cheeger corollaries (5.3, 5.6), the maxcut/bipartiteness spectral bounds, and the hypergraph bounds all lean on that identification or on similar hand-waved equivalences. Section 5.4, on manifolds, is worse: after deriving Corollaries 5.22–5.23, the text says \"We omit the detailed proofs of all the results in this section.\" For a paper whose selling point is the breadth of applications, that is a serious gap.\n\nThe self-citation pattern also deserves note. [38], [39], [40] are all the author's preprints, and the manuscript's new bounds rest on them without the lemmas being reproduced. That is not disqualifying by itself, but it makes the paper unverifiable as it stands.\n\nDo not desk-reject it. The abstract theory is a genuine candidate contribution, and the monotonicity inequality has the right shape to unify known Cheeger-type bounds. But it needs a referee with time to check the kernel reduction claim, and the author needs to either prove the identifications or clearly mark which corollaries depend on [40]. As is, it is a promising preprint, not a finished paper.\n\nRecommended action: send to a serious referee, with the explicit request to examine Example 5.1 and the omitted proofs in Section 5.4.","headline":"The core Choquet-extension theory is clean and novel, but the advertised applications lean on unproved identifications and the author's unpublished preprints; send to a serious referee but expect heavy revision.","tokens_in":35317,"tokens_out":2143,"would_cite":false,"duration_ms":19498,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Choquet-type extension theory yields a monotonicity inequality for global extension constants that unifies Cheeger, spectral gap, torsional rigidity, and isoperimetric bounds across graph limits, hypergraphs, Riemannian manifolds, and…","keywords":["Choquet extension","set-pair functions","global extension constants","monotonicity inequality","graph limits","hypergraphs","Riemannian manifolds","metric measure spaces"],"falsifier":"Compute both sides of equation (10) for $p=1$ on a simple two-block constant graphon: if the functional infimum over nonconstant functions differs from the set-based conductance $\\inf_S \\operatorname{cut}(S)/\\min(\\operatorname{vol}(S), \\operatorname{vol}(J\\setminus S))$, the identification in Example 5.1 fails. Alternatively, search finite graphs for a violation of the $p=1$, $r=2$ inequality $h^2/4 \\le \\lambda_2 \\le 2h$; any violation would disprove Theorem 5.2 and hence the monotonicity statement it is derived from.","tokens_in":34215,"feed_emoji":"📐","tokens_out":9902,"duration_ms":80998,"temperature":0.7,"pith_summary":"The paper builds a common analytic machine for quantities usually studied separately: conductance, spectral gaps, maxcut, bipartiteness ratio, torsional rigidity, and p-isoperimetric constants. It extends Choquet's integral from set functions to set-pair functions, integrates whole families of such extensions in L^p, and packages the result into global extension constants. The central result is a monotonicity inequality for these constants: under a natural relation between p, q, s, and t, the constant $c(\\Phi_p, J\\cdot K_q)$ is bounded by $t\\cdot c(\\Phi_{ps}, J\\cdot K_{qt})$, so that $p\\cdot c(\\Phi_p, J\\cdot K_p)$ grows with $p$. If correct, the inequalities recover and unify classical bounds and produce new ones on hypergraphs, graph limits, Riemannian manifolds, and metric measure spaces.","feed_headline":"One monotonicity law unifies Cheeger, spectral, torsional bounds","feed_subtitle":"Graph limits, hypergraphs, manifolds and metric measure spaces obey the same inequality.","key_machinery":"The central object is the global extension constant $c(\\Phi_p, J\\cdot K_q) = \\inf_{Y \\in \\mathcal{Y}} \\sup_{f \\in Y} \\Phi_p(f) / JfK_q$, where $\\Phi_p$ is the $L^p$ integral over a parameter space of Choquet extensions $b\\varphi_e(f)$ of a family of set functions, and $JfK_q$ is the $L^q$ norm of an intrinsic infinity norm $|f|_{\\varphi_e}$. The disjoint-pair version $\\Psi_p$ uses Choquet extensions $b\\psi_e(f)$ of set-pair functions, which handle signed structures. The proof of the monotonicity inequality runs through the Mazur map $f^r(x) = |f(x)|^r \\operatorname{sign}(f(x))$: the substitution turns an $L^p$ norm of $b\\varphi_e(f^t)$ into an $L^{ps}$ norm of $b\\varphi_e(f)$ times an $L^{qt}$ norm of $|f|$, and the admissible-set structure of the constants converts the pointwise inequality into the global one.","core_discovery":"On its own terms, the paper claims that a single family of constants controls a wide range of combinatorial and geometric parameters. The load-bearing theorem is the Extended Monotonicity Inequality: whenever $p, q, s, t \\ge 1$ satisfy $ps' = qt'$ for Hölder conjugates, one has $c(\\Phi_p, J\\cdot K_q) \\le t\\cdot c(\\Phi_{ps}, J\\cdot K_{qt})$, and $p\\cdot c(\\Phi_p, J\\cdot K_p)$ is nondecreasing in $p$. Under additional degree-related and concentration conditions, Theorem 4.12 gives the two-sided strengthening $(2C_\\Phi)^{1-t} c^t(\\Phi_{pt}, \\widehat{J\\cdot K}_{qt}) \\le c(\\Phi_p, \\widehat{J\\cdot K}_q) \\le t\\cdot c(\\Phi_{ps}, \\widehat{J\\cdot K}_{qt})$, and the same statements hold for the disjoint-pair version built from set-pair functions. The paper then identifies these constants with conductance, second Laplacian eigenvalues, maxcut, bipartiteness ratio, hypergraph expansion, torsional rigidity, and p-Laplacian eigenvalues, deriving the corresponding inequalities in its applications section.","pith_inferences":["A natural test is whether the constants $2$ and $t$ appearing in the bounds are sharp; the degree-related structure of Theorem 4.12 may admit refinements depending on the degree distribution of the graphon or hypergraph.","The functional reformulations in Section 3 make fractional set optimizations accessible to convex analysis, so one could construct minimizing sequences and algorithms for maxcut, bipartiteness ratio, and conductance directly from the Rayleigh-type quotients.","If the identification in Example 5.1 is accepted, the same monotonicity machinery should transfer to other limit objects, such as signed graphons and limits of delta matroids, since the disjoint-pair version is built for exactly those structures.","The paper leaves implicit the equality cases of Theorem 4.12; identifying the functions that realize the infimum would give extremal eigenfunctions for $p$-Laplacians across different values of $p$."],"forward_implications":["For $L^2$-graphons and graphings, the $p=1$, $r=2$ case gives $h_W^2/4 \\le \\lambda_2(L_W) \\le 2h_W$, covering the known Cheeger inequalities for graph limits.","For oriented hypergraphs, the machinery yields $2^{p-1} h^p / p^p \\le \\lambda_2(\\Delta_p) \\le 2^{p-1} h$, a Cheeger inequality for hypergraph $p$-Laplacians.","On compact Riemannian manifolds, the same inequality implies that $p\\cdot c_k(\\|\\nabla\\cdot\\|_p, \\|\\cdot\\|_p)$ increases with $p$, and that $T_p(M)\\lambda_p(M) \\le \\operatorname{vol}(M)^{p-1}$, a Pólya–Szegő type bound on torsional rigidity.","On metric measure spaces, the paper derives $p \\lambda_{k,p}(X)^{1/p} \\le q \\lambda_{k,q}(X)^{1/q}$ for $1 \\le p < q < \\infty$.","Spectral bounds for maxcut and bipartiteness ratio on graphons follow, including $4\\operatorname{MaxCut}(W) \\le \\|W\\|_1 \\lambda_{\\max}(L_W)$ and $\\beta_W^2 \\le 4 - \\lambda_{\\max}(L_W) \\le 4\\beta_W$."],"supporting_citations":[{"why":"Defines the original Choquet integral whose set-pair analog is introduced in Definition 2.1.","marker":"[10]"},{"why":"Introduces Choquet extension in submodular optimization, the starting point of Section 2.","marker":"[26]"},{"why":"Gives the Choquet extension for non-monotone submodular set functions whose analytic line is extended here.","marker":"[30]"},{"why":"Supplies the bounded-variation condition guaranteeing that Choquet integrals are well defined, used in Lemma 2.4 and Theorem 2.9.","marker":"[31]"},{"why":"Provides the kernel reduction lemma used in Example 5.1 to identify the constant $c_2(\\Phi_1, \\widehat{J\\cdot K}_1)$ with graphon conductance.","marker":"[40]"},{"why":"The Cheeger inequalities for graph limits whose graphon and graphing cases Corollaries 5.3 and 5.6 recover.","marker":"[21]"},{"why":"Supplies the oriented-hypergraph $p$-Laplacian and hyperedge expansion used in Theorem 5.14.","marker":"[19]"},{"why":"Supplies the coarea inequality and coarea formula used in Theorem 5.26 for metric measure spaces.","marker":"[12]"},{"why":"Supplies an elementary inequality used in the proof of the lower bound in Theorem 4.12.","marker":"[38]"}],"fun_headline_variants":["Monotonicity inequality unifies Cheeger and spectral bounds","One inequality links graph, manifold, and measure-space constants","A single law bounds conductance, rigidity, and eigenvalues"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing step is the identification of abstract global extension constants with concrete quantities: in Example 5.1 the intrinsic norm $|f|_{\\varphi_{xy}} = \\max(|f(x)|, |f(y)|)$ is replaced by the $p$-mean $((|f(x)|^p + |f(y)|^p)/2)^{1/p}$, and the resulting $c_2(\\Phi_1, \\widehat{J\\cdot K}_1)$ is declared to be the graphon conductance; if that replacement changes the infimum, the Cheeger and spectral corollaries do not follow from the theory.","fun_headline_variants_meta":{"raw":{"variants":["Monotonicity inequality unifies Cheeger and spectral bounds","One inequality links graph, manifold, and measure-space constants","A single law bounds conductance, rigidity, and eigenvalues"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000598,"raw_usage":{"total_tokens":2832,"prompt_tokens":1017,"completion_tokens":1815,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":633,"completion_tokens_details":{"reasoning_tokens":1770}},"tokens_in":633,"tokens_out":1815,"duration_ms":15302,"temperature":1.0,"reasoning_tokens":1770,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T14:11:44.395639+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute both sides of equation (10) for $p=1$ on a simple two-block constant graphon: if the functional infimum over nonconstant functions differs from the set-based conductance $\\inf_S \\operatorname{cut}(S)/\\min(\\operatorname{vol}(S), \\operatorname{vol}(J\\setminus S))$, the identification in Example 5.1 fails. Alternatively, search finite graphs for a violation of the $p=1$, $r=2$ inequality $h^2/4 \\le \\lambda_2 \\le 2h$; any violation would disprove Theorem 5.2 and hence the monotonicity statement it is derived from.","supporting_citations":[{"cited_title":"Theory of capacities.Ann","cited_arxiv_id":null,"evidence_quote":"Defines the original Choquet integral whose set-pair analog is introduced in Definition 2.1."},{"cited_title":"Submodular functions and convexity","cited_arxiv_id":null,"evidence_quote":"Introduces Choquet extension in submodular optimization, the starting point of Section 2."},{"cited_title":"Choquet extension of non-monotone submodular setfunctions.Acta Math","cited_arxiv_id":null,"evidence_quote":"Gives the Choquet extension for non-monotone submodular set functions whose analytic line is extended here."},{"cited_title":"Hidden critical and Morse equivalence behind duality: Theory and Applications","cited_arxiv_id":"2606.27004","evidence_quote":"Provides the kernel reduction lemma used in Example 5.1 to identify the constant $c_2(\\Phi_1, \\widehat{J\\cdot K}_1)$ with graphon conductance."},{"cited_title":"Cheeger inequalities for graph limits.Ann","cited_arxiv_id":null,"evidence_quote":"The Cheeger inequalities for graph limits whose graphon and graphing cases Corollaries 5.3 and 5.6 recover."},{"cited_title":"Cambridge Studies in Advanced Mathematics","cited_arxiv_id":null,"evidence_quote":"Supplies the oriented-hypergraph $p$-Laplacian and hyperedge expansion used in Theorem 5.14."},{"cited_title":"The equality case in Cheeger’s and Buser’s inequalities onRCDspaces.J","cited_arxiv_id":null,"evidence_quote":"Supplies the coarea inequality and coarea formula used in Theorem 5.26 for metric measure spaces."},{"cited_title":"Homological eigenvalues of graphp-Laplacians.J","cited_arxiv_id":null,"evidence_quote":"Supplies an elementary inequality used in the proof of the lower bound in Theorem 4.12."}],"review_version":1}