{"id":"fb3c1247-42a5-4b9d-bf8e-68e3f0b457c4","arxiv_id":"2507.17408","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A relative differential-escort transformation yields scale-invariant relative Fisher measures and sharp lower bounds by Kullback-Leibler and Rényi divergences with explicit minimizers.","lead":"This paper introduces a new transformation that reshapes a probability density relative to a reference density, and uses it to define new scale-invariant measures of dissimilarity. It proves sharp inequalities linking these measures to the Kullback-Leibler and Rényi divergences, with explicit minimizers.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.1(4.8) invokes the classical Stam inequality in a support regime where the transformed density fails the stated absolute-continuity hypothesis.","rationale":"The reader's weakest assumption correctly identifies the Stam inequality as the fragile step, but the precise failure mode is more specific: the classical Stam theorem (2.7) has an asymmetric hypothesis depending on the sign of 1+β−λ, and the transformed density F may have finite support with nonzero boundary values, so 'absolutely continuous on its support' does not imply the absolute continuity on R required by (2.7) when 1+β−λ>0. This is load-bearing because (4.8) is one of the two central sharp inequalities and is claimed for all pairs satisfying (2.1) plus the stated smoothness conditions. The concrete example shows the proof gap is not merely hypothetical. A secondary parameter-domain issue is that Theorem 4.1 does not impose ξ(λ,α)>0, although D_ξ is only defined for ξ>0 in Definition 2.1; this is fixable but should be added. The reader's conditional verdict remains appropriate: the paper contains a coherent framework and the inequalities are likely repairable, but the domain of validity of (4.8) needs to be either restricted or justified by an additional lemma. No change of verdict category is therefore recommended.","tokens_in":21734,"tokens_out":20431,"duration_ms":220618,"concrete_test":"Compute both sides of (4.8) for f(x)=2x, h(x)=1 on Ω=(0,1), p=2, β=λ=2, α=3/2, using the explicit transformed density F(y)=y^3 on (0,sqrt(2)) and the constant K^{(1)}_{2,2,2} from [42]. The zero extension of F is not absolutely continuous on R, so the proof of Theorem 4.1 cannot invoke (2.7) for this pair. If the numerical values show that (4.8) still holds, then a bounded-interval Stam lemma or an additional boundary-vanishing hypothesis is needed; if the inequality fails, Theorem 4.1 is false as stated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The principal gap concerns the Stam-like inequality (4.8), not the misprint in Lemma 3.2. The proof applies the classical triparametric Stam inequality (2.7) to F = R^{[h]}_α[f]. When 1+β−λ > 0, condition (2.7) is stated for densities on R that are absolutely continuous on R. But F is supported on (0, y_f), and y_f is finite whenever K_α[h||f] < ∞. For smooth densities f,h on a bounded common support with (f/h)(x_f) ≠ 0, F has a nonzero endpoint value at y_f, so its zero extension to R is not absolutely continuous. Example: take Ω=(0,1), f(x)=2x, h(x)=1, α=3/2. Then y'(x)=(2x)^{-1/2}, so y(x)=sqrt(2x) and F(y)=y^3 on (0,sqrt(2)), with F(sqrt(2))=2sqrt(2) > 0. For p=2, β=λ=2, the sign condition (2.6) holds and 1+β−λ=1>0, so (2.7) would require F to be absolutely continuous on R, which fails. The hypotheses listed in Remark 4.3 cover a C^1 ratio with compact support or a bounded derivative on unbounded support, but not this finite-support case. Hence (4.8) is not established for a natural class of pairs satisfying (2.1); the theorem's condition that R^{[h]}_α[f] be absolutely continuous on its support is insufficient as stated.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a 'relative differential-escort' transformation R_alpha^[h][f] depending on a reference density h (Definition 3.1), and uses it to define a biparametric relative Fisher divergence F_{p,lambda}[f||h] (Definition 4.1) and relative cumulative moments mu_{p,alpha}[f||h] (Definition 4.2). It proves that these new functionals are scale-invariant (Lemma 4.2) and transfers the classical moment-entropy and triparametric Stam inequalities to the relative setting (Theorem 4.1), with minimizers given by inverse-transformed stretched Gaussians. It also constructs adapted inequalities for a prescribed minimizer by solving for the reference density, leading to explicit formulas involving generalized trigonometric, hyperbolic, and incomplete gamma functions. The main algebraic steps are mostly sound, but the Stam-like inequality as stated has a hypothesis gap in the bounded-support case, and the theorem's parameter domain is not fully consistent with the paper's own definition of Rényi divergence.","tokens_in":22080,"tokens_out":14979,"duration_ms":145659,"significance":"The main conceptual contribution is a scale-invariant relative Fisher measure with sharp inequalities against Rényi and Kullback-Leibler divergences; this is a genuine improvement over earlier relative Fisher information measures (4.5) and (4.6), which scale nontrivially. The algebraic reductions in Lemmas 3.1, 3.2, 4.1 and 4.2 are sound, and the moment-entropy inequality (4.7) follows cleanly from the classical result (2.9). The paper also contains a constructive solution of the inverse minimizer problem, with explicit minimizers in several cases, which is a useful feature. The main caveat is that the Stam-like inequality (4.8) requires additional hypotheses in the bounded-support case, so the central theorem is currently over-stated.","major_comments":[{"comment":"The proof applies the classical triparametric Stam inequality (2.7) to F = R_alpha^[h][f]. When 1+beta-lambda > 0, (2.7) is stated for densities that are absolutely continuous on R, not merely on their support. However, F is supported on (0, y_f), and y_f is finite whenever K_alpha[h||f] < infinity (Remark 3.1). For a natural class of admissible pairs with bounded common support and (f/h)(x_f) != 0, the zero extension of F is discontinuous at y_f. For example, with Omega = (0,1), f(x) = 2x, h(x) = 1, alpha = 3/2, one gets y(x) = sqrt(2x) and F(y) = y^3 on (0, sqrt(2)), with F(sqrt(2)) = 2 sqrt(2) > 0; for p = 2, beta = lambda = 2, the sign condition (2.6) holds and 1+beta-lambda = 1 > 0. The condition 'R_alpha^[h][f] absolutely continuous' as stated, and the sufficient conditions in Remark 4.3, neither exclude nor cover this case. Thus (4.8) is not established for such pairs. The authors should either add the requirement that the zero extension is absolutely continuous on R (for example, vanishing endpoint values in the finite-support case), or prove a variant of (2.7) for densities on a bounded interval that accounts for boundary terms and verify that (4.8) holds in the present examples.","section":"§4.2, Theorem 4.1(4.8) and Remark 4.3"},{"comment":"The parameter domain of the inequalities is not consistent with the definition of Rényi divergence. Both (4.7) and (4.8) contain exp(D_xi[f||h]) with xi(lambda, alpha) = 1 + alpha(lambda - 1), but Definition 2.1 defines D_xi only for xi > 0. The hypotheses alpha > 0 and lambda > 1/(1+p*) do not imply xi > 0; for instance, p* = 1, lambda = 0.9, alpha = 20 gives xi = -1. The theorem should either impose alpha(1 - lambda) < 1 when lambda < 1, or explicitly state that the expressions are interpreted via K_xi^{1/(xi-1)} for all real xi for which the integral is finite. As written, part of the stated parameter range is not covered by the paper's own definitions.","section":"§4.2, Theorem 4.1 (parameter range)"},{"comment":"The sharpness claim of Theorem 4.1 depends on the finiteness of N_{1/alpha*}[g_{p,lambda}] and N_{1/alpha*}[g_{p,beta,lambda}], which is needed for Definition 3.2, but the paper gives no characterization of this condition and the normalization constants a_{p,lambda} are not given. Consequently the reader cannot determine for which (p, lambda, alpha) the proposed minimizers are admissible, and the explicit formulas in Section 4.3 (for example (4.13)-(4.16)) silently assume this finiteness. A parameter analysis, or at least explicit sufficient conditions for the finiteness of these Rényi entropy powers, should be supplied before the sharpness statement of Theorem 4.1 can be considered complete.","section":"§4.2 and §4.3 (sharpness and minimizers)"}],"minor_comments":[{"comment":"The statement writes sigma_{p,lambda} for the relative cumulative moment, while Definition 4.2 defines sigma_{p,alpha}; the subscript should be alpha throughout.","section":"§4.1, Lemma 4.2"},{"comment":"The references to inequalities (4.18) and (4.19) appear before those equations are defined in Section 4.4; they should refer to (4.7) and (4.8).","section":"§4.3"},{"comment":"The notation N_lambda^{1-lambda} in (3.12) is easy to misread; adding a parenthetical or spacing would clarify that it means (N_lambda)^{1-lambda}, as used in the proof and in (4.9).","section":"§3.1, Lemma 3.2"},{"comment":"In the exponential example, the intermediate expression for y(x) contains a cumbersome term of the form a^{alpha-1}(a alpha - a - alpha) that should be simplified or checked; the final support condition alpha < a* is clear, but the derivation would benefit from an extra step.","section":"§3.2"}],"recommendation":"major_revision","confidential_remarks":"The paper relies heavily on the authors' own previous work (notably [40], [42], [44], [45]), but the new transformation and the scale-invariant functionals are sufficiently distinct to be of interest. The correctness issues are technical rather than conceptual, and I expect they can be fixed within the manuscript's scope. I would advise the editor to send the revision back to the same referee, because verifying the repaired Stam-inequality hypotheses requires checking both the classical inequality's hypotheses and the endpoint behavior of the transformed density."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe short version: this paper introduces genuinely new objects: a relative differential-escort transformation, a biparametric relative Fisher divergence that is invariant under joint scaling, and relative cumulative moments. It derives sharp moment-entropy and Stam-like inequalities with explicit minimizers. The derivations are mostly transparent and honest: the new inequalities come from applying classical moment-entropy and Stam inequalities to the transformed density, not from assuming the conclusion. The scaling invariance is a real improvement over earlier relative Fisher measures, which pick up an r^2 factor under scaling.\n\nWhere it gets soft: first, Lemma 3.2 as typeset is wrong; the identity should involve N_lambda^{1-lambda} on the left, not what is printed. The normalization constants a_{p,lambda} are left as \"complicated,\" which is annoying but minor. Second, and more substantive, the Stam-like inequality (4.8) has a hypothesis gap. The proof applies the classical triparametric Stam inequality to F = R^{[h]}_alpha[f]. In the regime 1+beta-lambda > 0, the classical statement requires F to be absolutely continuous on R. But when the common support is bounded and y_f is finite, F has compact support and generically a nonzero endpoint value, so its zero extension jumps. Concrete example: Omega=(0,1), f(x)=2x, h(x)=1, alpha=3/2 gives F(y)=y^3 on (0,sqrt(2)), with F(sqrt(2))=2sqrt(2)>0; with p=2, beta=lambda=2 the sign condition holds. The zero extension is not absolutely continuous on R. The theorem's condition merely says R^{[h]}_alpha[f] is absolutely continuous, and Remark 4.3's sufficient conditions do not cover this bounded-support case. So (4.8) is not established for a natural class of pairs satisfying (2.1). This is a real gap, not cosmetic, though it may be patchable by invoking a compact-support version of Stam or by adding endpoint conditions. The reader's stress test is on target here.\n\nEverything else holds up. No circularity; the cited Stam result in [44] traces to earlier independent work. The sharpness statements depend on a finiteness assumption for N_{1/alpha*}[g_{p,lambda}] that is not parameter-analyzed in detail; that is a moderate caveat, not a flaw.\n\nWho this is for: people working on informational inequalities, relative Fisher information, and generalized complexity measures. They will want this framework. I would send it to a serious referee: the core idea is good, and the main gap is identifiable and probably fixable. After the hypothesis in (4.8) is repaired and the typos cleaned up, publication is reasonable.","headline":"Worth engaging: the new scale-invariant relative Fisher and cumulative measures are genuinely new and mostly proven, but the Stam-like inequality has a real absolute-continuity gap on bounded supports that a referee should insist be patched.","tokens_in":22646,"tokens_out":4741,"would_cite":false,"duration_ms":49855,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["94A17","26D15","60E15"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper introduces a relative differential-escort transformation and uses it to prove sharp lower bounds, in terms of Kullback-Leibler and Rényi divergences, for two new families of scaling-invariant relative Fisher and…","keywords":["relative Fisher divergence","Rényi divergence","Kullback-Leibler divergence","relative differential-escort transformation","relative cumulative moment","stretched Gaussian","generalized trigonometric functions","Stam inequality"],"falsifier":"Take the reference $h$ to be a Gaussian and pick $f$ such that $f/h$ is smooth with unbounded derivative on an unbounded support; if the left side of (4.8) can be driven below the claimed constant $\\alpha^{(\\lambda-\\beta-1)/(\\alpha\\beta)}(K^{(1)}_{p,\\beta,\\lambda})^{1/\\alpha}$, then the absolute-continuity hypothesis is genuinely needed. Alternatively, for $h$ exponential and $f$ equal to the claimed minimizer $R^{-1,[h]}_\\alpha[g_{p,\\lambda}]$, evaluate both sides of (4.7) numerically; equality to machine precision would confirm the sharpness computation.","tokens_in":21483,"feed_emoji":"📐","tokens_out":8580,"duration_ms":80223,"temperature":0.7,"pith_summary":"The paper's aim is to build a relative version of the differential-escort transformation, a change of variables plus rescaling of a density by a reference density $h$, and to use it to define two new biparametric families of informational measures: a relative Fisher divergence and a relative cumulative moment. The central achievement it asserts is that these measures are invariant under scaling of both densities, unlike previously known relative Fisher measures, and that they satisfy sharp lower bounds controlled by the Kullback-Leibler and Rényi divergences. Explicit minimizers are given as inverse relative differential-escort transforms of stretched Gaussians, and for a prescribed minimizer the reference density takes a closed form built from generalized trigonometric functions. The authors see this as transferring the classical moment-entropy and Stam inequalities from the single-density setting to the relative framework.","feed_headline":"Scale-invariant relative Fisher measures obey sharp divergence bounds","feed_subtitle":"Explicit minimizers are inverse escort transforms of stretched Gaussians, tied to generalized trigonometric functions.","key_machinery":"The engine of the paper is the relative differential-escort transformation $R^{[h]}_\\alpha[f](y) = (f(x)/h(x))^\\alpha$, with the new coordinate defined by $y'(x)=f(x)^{1-\\alpha}h(x)^\\alpha$ (Definition 3.1). It generalizes the differential-escort transformation: choosing the reference density $h$ to be uniform on the support recovers the standard case. The transformation is a bijection between densities when the Rényi entropy power $N_{1/\\alpha^*}[g]$ is finite, with inverse $R^{-1,[h]}_\\alpha[g]$ built by scaling $g$ by its Rényi entropy power. The mechanism of the proofs is to compute, in terms of the original pair $(f,h)$, the Rényi entropy power (Lemma 3.2), the $(p,\\lambda)$-Fisher information (Lemma 4.1) and the $p$-th cumulative moment of the transformed density, so that the classical inequalities (2.9) and (2.7) become the new sharp relative inequalities.","core_discovery":"The central discovery is that the relative differential-escort transformation $R^{[h]}_\\alpha[f](y)=(f(x)/h(x))^\\alpha$ with $y'=f^{1-\\alpha}h^\\alpha$ converts a classical inequality for a single density into a sharp inequality for a pair of densities. Applying the classical moment-entropy inequality to $R^{[h]}_\\alpha[f]$ yields $e^{D_{\\xi(\\lambda,\\alpha)}[f\\|h]}\\sigma_{p^*,\\alpha}[f\\|h] \\geq (K^{(0)}_{p,\\lambda})^{1/\\alpha}$ (Theorem 4.1), and applying the triparametric Stam inequality yields the companion bound $(e^{-D_{\\xi(\\lambda,\\alpha)}[f\\|h]}\\phi_{p,\\beta\\alpha}[f\\|h])^{1+\\beta-\\lambda} \\geq \\alpha^{(\\lambda-\\beta-1)/(\\alpha\\beta)}(K^{(1)}_{p,\\beta,\\lambda})^{1/\\alpha}$; in both cases equality is attained by the inverse transform of the stretched Gaussian minimizers $g_{p,\\lambda}$ and $g_{p,\\beta,\\lambda}$. Because the new relative Fisher divergence and relative cumulative moment are invariant under simultaneous scaling, the bounds are not an artifact of units, and the $\\lambda\\to 1$ case yields clean inequalities involving the Shannon entropy and the Kullback-Leibler divergence.","pith_inferences":["Beyond the paper: choosing $h$ to be a physically motivated reference density (for example a Gaussian or a known ground-state density) makes the sharp minimizers a parameter-free family of model densities; comparing empirical densities against these minimizers could provide a direct estimator of the nonextensivity parameter $\\lambda$.","Beyond the paper: the construction is coordinate-based rather than dimension-specific, so the same transformation should produce scaling-invariant relative Fisher and cumulative-moment measures in higher dimensions; verifying the analogues of (4.7)-(4.8) there would be a direct test of the framework.","Beyond the paper: the invertibility of $R^{[h]}_1$ connects the sequence of Rényi divergences of $f$ against $h$ to the Rényi entropy powers of the transformed density, suggesting a relative Hausdorff moment problem in which divergence data can recover $f$ by inverting the transformation."],"forward_implications":["For any fixed reference density $h$, inequalities (4.7) and (4.8) give explicit, sharp lower bounds on the relative cumulative moment and relative Fisher divergence in terms of Rényi divergences, with the minimizer written down in closed form via the inverse transformation.","In the limit $\\lambda\\to 1$ the bounds become simple inequalities involving the Shannon entropy and the Kullback-Leibler divergence, and when $\\beta=\\lambda$ the two inequalities multiply to a Cramér-Rao-like bound in the relative setting.","The relative Fisher-Shannon complexity measure defined in (4.20) is monotone under Gaussian convolution (Proposition 4.1), extending a classical monotonicity property to this relative framework.","Fixing a desired minimizer $f_*$ and solving for the reference density produces adapted measures whose two factors are each minimized by the reference density $h_*$ while the product of the factors is minimized exactly at $f_*$, a separation of roles not seen in the classical inequalities.","Because both new measures are invariant under simultaneous scaling of $f$ and $h$, the inequalities remain valid under arbitrary changes of units, which the previous relative Fisher measures (4.5) and (4.6) did not enjoy."],"supporting_citations":[{"why":"Supplies the classical moment-entropy inequality (2.9) with stretched-Gaussian minimizers from which (4.7) is derived.","marker":"[10]"},{"why":"Provides the (β,q)-generalized Fisher information framework and q-Gaussian minimizers that ground the generalized Stam constants used in (4.8).","marker":"[11]"},{"why":"Defines cumulative moments and the notation for the sharp constants K^{(0)}_{p,λ} and K^{(1)}_{p,β,λ}, and connects generalized trigonometric functions to the minimizers.","marker":"[40]"},{"why":"Establishes the monotonicity of the Fisher-Shannon complexity under Gaussian convolution, which Proposition 4.1 extends to the relative setting.","marker":"[41]"},{"why":"Establishes the extended triparametric Stam inequality (2.7) that, applied to the transformed density, yields (4.8).","marker":"[42]"},{"why":"Develops the differential-escort technique and the explicit constants and minimizers that Theorem 4.1 takes as its starting point.","marker":"[44]"},{"why":"Introduces the generalized sine and cosine functions that appear in the explicit form of the minimizers and adapted measures.","marker":"[47]"},{"why":"Links the (p,q)-generalized trigonometric functions to a p-Laplacian eigenvalue problem, the source of the two-parameter sine and cosine used in Section 5.","marker":"[48]"},{"why":"Supplies the moment-entropy inequality for Rényi entropy whose limit gives the λ=1 case (4.10).","marker":"[53]"},{"why":"Provides one of the previous relative Fisher measures (4.5) that the paper shows is not scale-invariant, motivating the new definition.","marker":"[55]"}],"fun_headline_variants":["Scale-invariant relative Fisher measures sharpen divergence bounds","Relative escort transforms yield sharp KL-Renyi inequalities","Optimal bounds connect relative Fisher and KL-Renyi divergences","Scale-free Fisher measures give sharp KL and Renyi divergence bounds","New relative Fisher measures achieve sharp divergence inequalities"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument inherits the classical Stam inequality (2.7), which requires the parameter sign condition (2.6) and absolute continuity of the transformed density $R^{[h]}_\\alpha[f]$; the admissible pairs $(f,h)$ satisfying these conditions are not fully characterized, and the sharpness claim also assumes $N_{1/\\alpha^*}[g_{p,\\lambda}]$ is finite.","fun_headline_variants_meta":{"raw":{"variants":["Scale-invariant relative Fisher measures sharpen divergence bounds","Relative escort transforms yield sharp KL-Renyi inequalities","Optimal bounds connect relative Fisher and KL-Renyi divergences","Scale-free Fisher measures give sharp KL and Renyi divergence bounds","New relative Fisher measures achieve sharp divergence inequalities"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000247,"raw_usage":{"total_tokens":1582,"prompt_tokens":1022,"completion_tokens":560,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":638,"completion_tokens_details":{"reasoning_tokens":482}},"tokens_in":638,"tokens_out":560,"duration_ms":20398,"temperature":1.0,"reasoning_tokens":482,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T14:50:56.646857+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the reference $h$ to be a Gaussian and pick $f$ such that $f/h$ is smooth with unbounded derivative on an unbounded support; if the left side of (4.8) can be driven below the claimed constant $\\alpha^{(\\lambda-\\beta-1)/(\\alpha\\beta)}(K^{(1)}_{p,\\beta,\\lambda})^{1/\\alpha}$, then the absolute-continuity hypothesis is genuinely needed. Alternatively, for $h$ exponential and $f$ equal to the claimed minimizer $R^{-1,[h]}_\\alpha[g_{p,\\lambda}]$, evaluate both sides of (4.7) numerically; equality to machine precision would confirm the sharpness computation.","supporting_citations":[{"cited_title":"Lutwak, D","cited_arxiv_id":null,"evidence_quote":"Supplies the classical moment-entropy inequality (2.9) with stretched-Gaussian minimizers from which (4.7) is derived."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the (β,q)-generalized Fisher information framework and q-Gaussian minimizers that ground the generalized Stam constants used in (4.8)."},{"cited_title":"Puertas-Centeno and S","cited_arxiv_id":null,"evidence_quote":"Defines cumulative moments and the notation for the sharp constants K^{(0)}_{p,λ} and K^{(1)}_{p,β,λ}, and connects generalized trigonometric functions to the minimizers."},{"cited_title":"Rudnicki, E","cited_arxiv_id":null,"evidence_quote":"Establishes the monotonicity of the Fisher-Shannon complexity under Gaussian convolution, which Proposition 4.1 extends to the relative setting."},{"cited_title":"Zozor, D","cited_arxiv_id":null,"evidence_quote":"Establishes the extended triparametric Stam inequality (2.7) that, applied to the transformed density, yields (4.8)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Develops the differential-escort technique and the explicit constants and minimizers that Theorem 4.1 takes as its starting point."},{"cited_title":"Lindqvist","cited_arxiv_id":null,"evidence_quote":"Introduces the generalized sine and cosine functions that appear in the explicit form of the minimizers and adapted measures."},{"cited_title":"Dr´ abek and R","cited_arxiv_id":null,"evidence_quote":"Links the (p,q)-generalized trigonometric functions to a p-Laplacian eigenvalue problem, the source of the two-parameter sine and cosine used in Section 5."},{"cited_title":"Lutwak, D","cited_arxiv_id":null,"evidence_quote":"Supplies the moment-entropy inequality for Rényi entropy whose limit gives the λ=1 case (4.10)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides one of the previous relative Fisher measures (4.5) that the paper shows is not scale-invariant, motivating the new definition."}],"review_version":1}