Old and New on Strongly Subadditive/Superadditive Functions

Constantin P. Niculescu

REVIEW 2 major objections 4 minor 1 cited by

Old and New on Strongly Subadditive/Superadditive Functions

T0 review · 2 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A monotone strongly superadditive function on a convex cone, composed with any nondecreasing convex function, yields a two-term inequality that symmetrizes to a Popoviciu-type inequality.

desk verdict Theorem 8 is a genuine and correct new inequality, but the paper as posted has two concrete internal errors—Corollary 3 and Corollary 7—that make it unsafe as a reference until fixed. read the letter →

arxiv 2501.13695 v1 pith:MXMZTIFC submitted 2025-01-23 math.FA math.CA

Constantin P. Niculescu This is my paper · ORCID

classification math.FAmath.CA MSC 26A5139B6246B2026D15

keywords stronglysubadditivefunctionssuperadditiveconvexconesweakmajorizationPopoviciuinequalitycompletelymonotonedeterminantvonNeumannentropy

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Strongly subadditive and superadditive functions, whose second-order differences have a definite sign on every triple in a convex cone, are far less studied than ordinary subadditive and superadditive functions, even though the class includes von Neumann entropy, the determinant on positive definite matrices, and trace powers. The paper maps this territory by characterizing these functions in one variable, in several variables via Hessian signs, and through their connection to submodularity and completely monotone functions. Its main new result, Theorem 8, says that a monotone strongly superadditive function, pushed through any nondecreasing convex function, gives a three-variable inequality of Popoviciu type. If correct, this turns a wide stock of known examples into generators of explicit inequalities in matrix analysis, probability, and lattice theory.

What carries the argument

The load-bearing machinery is the weak majorization theorem of Tomi\'c and Weyl: for a nondecreasing convex $f$, when a decreasing sequence of numbers has partial sums bounded above by another sequence, the sum of $f$ over the first sequence is bounded above by the sum over the second. Theorem 8 feeds this theorem the four numbers $\Phi(y+z)$, $\Phi(x+z)$, $\Phi(x+y+z)$, and $\Phi(z)$; strong superadditivity supplies the total-sum comparison, and monotonicity supplies the ordering needed to apply weak majorization. Around this, the paper uses second-order difference operators $\Delta_x\Delta_y\Phi(z)$ to characterize strong subadditivity and superadditivity, and uses the Laplace-transform representation of completely monotone functions to generate examples that satisfy the hypotheses.

What would settle it

Check whether monotonicity is really needed: let $C=\mathbb{R}_+^2$, $\Phi(x_1,x_2)=x_1^2+x_2^2-2x_1-2x_2$, $f(t)=\max(t,0)$, $x=(1+\sqrt{3/2},0)$, $y=(0,1)$, and $z=(0,0)$. The Hessian of $\Phi$ has nonnegative entries and $\Phi(0)=0$, so $\Phi$ is strongly superadditive, yet $\Phi$ is not monotone. The claimed inequality becomes $0\ge 1/2$, which is false.

Watch

Extended reading notes

Core claim

The paper's central assertion is Theorem 8: if $C$ is a convex cone, $\Phi:C\to[0,\infty)$ is strongly superadditive, meaning $\Phi(x+y+z)+\Phi(z)\ge \Phi(x+z)+\Phi(y+z)$ for all $x,y,z\in C$, and $\Phi$ is monotone in the cone order, then for every nondecreasing convex function $f$ and all $x,y,z\in C$, $f(\Phi(x+y+z))+f(\Phi(z))\ge f(\Phi(x+z))+f(\Phi(y+z))$. Symmetrizing this two-term inequality yields a Popoviciu-type inequality. The author presents this as a new functional-inequality principle, with the determinant, trace functions, $L^p$ norms, and shifted completely monotone functions as instances.

Load-bearing premise

The load-bearing premise is that $\Phi$ is monotone in the cone order ($x\le y$ implies $\Phi(x)\le\Phi(y)$); without it, the entries fed into weak majorization cannot be ordered, and the claimed inequality can fail even for strongly superadditive functions.

Editorial extensions

If this is right

Corollary 7: for all $A,B,C\in\mathrm{Sym}_+(N,\mathbb{R})$ and $p\ge 1$, $(\det A)^p+(\det C)^p+\det(A+B+C)^p \ge \frac{2}{3}\left[(\det(A+B))^p+(\det(B+C))^p+(\det(A+C))^p\right]$.
For $p\in[1,2]$, $\mathrm{trace}(A^p)$ is strongly superadditive on positive semidefinite matrices, and for $p\in[0,1]$ it is strongly subadditive, so Theorem 8 produces matching functional inequalities for both ranges.
For $p>1$, the function $f\mapsto \|f\|_p^p$ is strongly superadditive on the positive cone of $L^p(\mathbb{R})$, giving Banach-lattice inequalities.
The log-sum-exp function is not strongly superadditive in general but is comonotonic strongly superadditive, so the theorem's inequalities hold for pairwise comonotonic triples, as noted in Remark 5.
Completely monotone functions, shifted and centered at their origin value, become strongly superadditive; applying Theorem 8 to them yields explicit inequalities for functions such as $(\det(I+A))^{-\beta}-1$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editor's inference: The theorem can be read as an inequality machine: each monotone strongly superadditive $\Phi$ and each nondecreasing convex $f$ on its range gives valid two- and three-term inequalities, so choosing $f(t)=e^t$, $f(t)=t^p$, or $f(t)=\max(t,0)$ expands the paper's determinant and trace corollaries into families not listed there.
Editor's inference: In the one-dimensional case, convexity plus $\Phi(0)\le 0$ automatically makes $\Phi$ strongly superadditive and renders the monotonicity condition redundant, so the genuinely new content of Theorem 8 lies in cones of dimension two or higher, where checking monotonicity via the differential (as in Theorem 3) is the practical entry point.
Editor's inference: The combination of strong superadditivity and monotonicity behaves like a discrete convexity in the cone order, and the paper's link to submodularity suggests that Popoviciu-type inequalities may also hold for monotone submodular set functions or Choquet integrals, where monotonicity is often built into the model.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You should know two things. First, Theorem 8 is real: the Popoviciu-type inequality for monotone strongly superadditive functions is new, and the proof via Tomić–Weyl weak majorization is correct. The comonotonic log-sum-exp observation in Remark 4 is also new and looks right. Second, the paper contains two concrete false statements. Corollary 3 claims log det is strongly subadditive, which is false under the paper's own definition because log det is not subadditive (Remark 3 admits this). Corollary 7, the advertised determinant application of Theorem 8, is false as written: for N=1, A=C=0, B=10, p=1, it asserts 10 ≥ 40/3. The stress-test note is right. The symmetrized Theorem 8 gives F(A)+F(B)+F(C)+3F(A+B+C) ≥ 2(F(A+B)+F(B+C)+F(A+C)), not the displayed inequality. So the headline application is not a consequence of Theorem 8.

What the paper does well: it is a readable survey of strongly sub/superadditive functions, with a clean catalogue of examples (trace powers, determinant, Shannon entropy, complete monotone functions) and several correct characterizations (Theorems 1–4, Corollaries 1–2, 4–6). The proofs are short and mostly self-contained. Theorem 8, once separated from Corollary 7, is a legitimate extension of the majorization program and could be useful.

Soft spots beyond the false statements: Theorem 3 has a domain ambiguity—it states Φ on E++ but uses Φ(0) and strong additivity on the closed cone. The monotonicity assumption in Theorem 8 is real and not discussed; without it the weak majorization step fails. The paper also leans heavily on the author's own textbook [20], though the cited background claims do not seem circular.

Who this is for: anyone working on functional inequalities, matrix analysis, or entropy-like functions will find Theorem 8 and the catalogue valuable. But the paper is not ready as a reference in its current form. The false corollaries need correcting, and the determinant application should be either fixed or withdrawn. I would send it to review because the core idea is sound and new, but I would expect a revision.

Recommendation: reject from desk; send to referee with a clear request to check Corollaries 3 and 7 and the domain issue in Theorem 3.

Referee Report

2 major / 4 minor

Summary. The paper investigates strongly subadditive and strongly superadditive functions on convex cones. It gives one-variable characterizations (Theorems 1 and 2), Hessian characterizations in several variables (Theorem 4), a connection to submodularity, and constructions from complete monotone functions. Its main new result is Theorem 8, which uses the Tomić–Weyl weak majorization theorem to generate Popoviciu-type inequalities from monotone strongly superadditive functions. The paper also states a number of examples and applications, including determinant inequalities.

Significance. The core theoretical contribution, Theorem 8, appears correct and provides a clean mechanism for generating two- and three-term functional inequalities from monotone strongly superadditive functions. The paper also collects a useful set of examples, several with proofs, such as Corollaries 1, 2, and 4. However, the manuscript contains two significant errors in its advertised applications: Corollary 7 is numerically false, and Corollary 3 is inconsistent with the paper's own definition of strong subadditivity and with Remark 3. Because these are load-bearing claims, the manuscript requires major revision before it can be considered for publication.

major comments (2)

[§5, Corollary 7] Corollary 7 is false as stated. For N=1, Sym+(1,R)=[0,∞) and det(X)=X. Taking p=1, A=0, B=10, and C=0, the claimed inequality becomes 0+0+10 ≥ (2/3)(10+10+0)=40/3, which is false. Moreover, this inequality does not follow from the symmetrized Popoviciu inequality in Theorem 8: applying that theorem with Φ=det and f(t)=t^p yields (det A)^p+(det B)^p+(det C)^p+3(det(A+B+C))^p ≥ 2[(det(A+B))^p+(det(B+C))^p+(det(A+C))^p], which contains an extra (det B)^p term and a factor 3 on the final determinant term. The corollary should be corrected to match the actual consequence of Theorem 8, or withdrawn.
[§2, Corollary 3] Corollary 3 is inconsistent with the paper's definition of strong subadditivity. In Section 1, a function is called strongly subadditive only if it also satisfies the subadditivity inequality (1.1). For N=1 and A1=1, Φ(x)=log x is not subadditive (e.g., Φ(1+1)=log 2 > 0 = Φ(1)+Φ(1)), yet Corollary 3 asserts that Φ is strongly subadditive. This contradiction is explicitly acknowledged in Remark 3, which states that log det is neither subadditive nor superadditive but satisfies the 2-monotone inequality (1.2). Corollary 3 should be reworded to assert only (1.2), or the definition of strong subadditivity should be revised consistently throughout the paper.

minor comments (4)

[§5, Theorem 7] The statement of Theorem 7 is ambiguous: the phrase "while when b1 ≤ ... ≤ bn, then the conclusion works in the reverse direction" is imprecise regarding the required ordering of the a's and the direction of the inequality. The proof uses only the first clause, so the statement could be clarified by stating the standard weak majorization theorem in full.
[§2, Proof of Corollary 2] There is an extra parenthesis in the line "Φ is differentiable on Sym++(N,R))", which should read "Sym++(N,R)".
[§2, Theorem 3] Theorem 3 states that Φ is defined on the interior cone E++ but then refers to Φ(0); the paper should clarify that Φ is defined on the closure of the positive cone or that Φ(0) is interpreted as the value at the origin when the origin is in the domain.
[§4] The sentence "The same works for the function Φβ" is vague about the precise set of exponents for which the claimed strong superadditivity holds; a more explicit statement would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's derivations rely on external theorems and explicit verification, not on self-referential inputs.

full rationale

The paper's main new result, Theorem 8, derives a Popoviciu-type functional inequality from the assumption of strong superadditivity plus monotonicity of the function Phi, using the classical Tomić–Weyl weak majorization theorem. The proof explicitly verifies the majorization conditions a1 <= b1 and a1+a2 <= b1+b2 from the hypotheses, so the conclusion follows directly from an external, well-established result. No parameter is fitted, no definition implicitly contains the claimed conclusion, and no load-bearing step cites the author's own work in place of proof. The characterizations in Theorems 1–4 are proven from standard convex analysis facts (Hardy–Littlewood–Pólya majorization, Jensen's midpoint convexity theorem, differentiation criteria, and the Bernstein–Hausdorff–Widder–Choquet representation). Citations to the author's own book [20] appear only for background results such as the majorization theorem and the differential of the Lp norm; these are standard and independently verifiable, not unique to the author. Similarly, the determinant inequality Corollary 7 is grounded in the independently proven strong superadditivity of det (Corollary 2), which is established via Theorem 3 and Weyl's monotonicity principle. Even if Corollary 7 contains a potential algebraic error (as a skeptical correctness note might suggest), that is a matter of mathematical validity, not circularity. The chain of reasoning is transparent and self-contained against external benchmarks; therefore the circularity score is 0.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No free parameters or invented entities appear. The paper is purely analytical and relies on standard theorems, though a few internal sign and domain errors need correction.

assumptions (7)

standard math Tomić-Weyl weak majorization theorem (Theorem 7 of the paper)
Used as the black box in the proof of Theorem 8 to pass from partial sum inequalities to f-sum inequalities.
standard math Hardy-Littlewood-Pólya majorization theorem
Used in Lemma 1 and several examples to identify convexity via second differences.
standard math Jensen's theorem: midpoint convex plus continuity implies convex
Used in Lemma 1 to go from nonnegative second differences to convexity.
standard math Weyl's monotonicity principle for eigenvalues
Used in the proof of strong superadditivity of det (Corollary 2).
standard math Bernstein-Hausdorff-Widder-Choquet representation theorem for completely monotone functions on cones
Used in Section 4 to prove that completely monotone functions are 2-monotone increasing.
standard math Löwner-Heinz operator monotonicity theorem
Used to establish trace function examples such as trace A^p.
domain assumption Ordered Banach spaces satisfy (OBS1) generating cone and (OBS2) monotone norm
The framework in the appendix; Theorem 3 is stated for such spaces.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Old and New on Strongly Subadditive/Superadditive Functions." pith.science (2026). https://pith.science/paper/MXMZTIFC

@misc{pith2026250113695,
  author       = {Pith},
  title        = {Pith review of: Old and New on Strongly Subadditive/Superadditive Functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MXMZTIFC}},
  note         = {Machine review of arXiv:2501.13695}
}

read the original abstract

In this paper we provide insight into the classes of strongly subadditive/superadditive functions by highlighting numerous new examples and new results.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

The model theory of metric lattices: pseudofinite partition lattices
math.CO 2025-07 conditional novelty 7.0 of 10

Infinite pseudofinite limits of partition lattices have a definable Boolean sublattice of modular elements, and each element is determined by its set of modular selectors.

Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages · cited by 1 Pith paper

[20]

A Contemporary Approach

Niculescu, C.P., Persson, L.-E.: Convex Functions and Their Applications. A Contemporary Approach. 3rd Ed., CMS Books in Mathematics Vol. 23, Springer-Verlag, New York (2025)

work page 2025
[1]

Graduate Studies in Mathem atics vol

Aliprantis, C.D.[ ?], Tourky, R.: Cones and Duality. Graduate Studies in Mathem atics vol. 84, American Mathematical Society, Providence, R.I. (2007)

work page 2007
[2]

Amann, H.: Multiple positive ﬁxed points of asymptotica lly linear maps. J. Functional Anal- ysis 17(2), 174-213 (1974)

work page 1974
[3]

Acta Mathematica Hun- garica 128(4), 386–394 (2010)

Audenaert, K., Hiai, F., Petz, D.: Strongly subadditive functions. Acta Mathematica Hun- garica 128(4), 386–394 (2010)

work page 2010
[4]

In: International Conferen ce on Engineering Optimization (EngOpt 2008), Rio de Janeiro, June 1–5, 2008, Paper # 0615, P rinted and CDROM Pro- ceedings, ISBN 978-85-7650-152-7

Bouhtou, M., Gaubert, S., Sagnol, G.: Optimization of ne twork traﬃc measurement: a semi- deﬁnite programming approach. In: International Conferen ce on Engineering Optimization (EngOpt 2008), Rio de Janeiro, June 1–5, 2008, Paper # 0615, P rinted and CDROM Pro- ceedings, ISBN 978-85-7650-152-7

work page 2008
[5]

Cambr idge University Press, Cambridge, UK (2004)

Boyd, S., Vandenberghe, L.: Convex Optimization. Cambr idge University Press, Cambridge, UK (2004)

work page 2004
[6]

Contemporary Mathematics 529, 73–140 (2010)

Carlen, E.A.: Trace inequalities and quantum entropy: A n intro- ductory course. Contemporary Mathematics 529, 73–140 (2010). URL http://www.ueltschi.org/AZschool/notes/EricCarlen.pdf

work page 2010
[7]

Choquet, G.: Deux exemples classiques de repr´ esentati on int´ egrale. Enseign. Math. 15(2), 63-75 (1969)

work page 1969

Show all 28 references

[8]

Kluw er Academic Publisher, Dordrecht (1994)

Denneberg, D.: Non-Additive Measure and Integral. Kluw er Academic Publisher, Dordrecht (1994)

1994
[9]

Springer, New York (2011)

Dudley, R.M., Norvaisa, R.: Concrete functional calcul us. Springer, New York (2011)

2011
[10]

Aequationes Math

Gal, S.G., Niculescu, C.P.: A note on the Choquet type op erators. Aequationes Math. 95(3), 433-447 (2021)

2021
[11]

Positivity 28(2), article 14 (2024)

Gal, S.G., Niculescu, C.P.: Nonlinear operator extens ions of Korovkin’s theorems. Positivity 28(2), article 14 (2024)

2024
[12]

Hansen, F.: Trace functions with applications in quant um physics. J. Stat. Phys. 154, 807– 818 (2014)

2014
[13]

Volume II of London Math Society Lecture Notes Series, pages 84–114

Kozhasov, K., Michalek, M., Sturmfels, B.: Positivity certiﬁcates via integral representations. Volume II of London Math Society Lecture Notes Series, pages 84–114. Cambridge University Press (2022). OLD AND NEW ON STRONGLY SUBADDITIVE/SUPERADDITIVE FUNCTIO NS 17

2022
[14]

IEEE Trans

Lami, L., Hirche, C., Adesso, G., Winter, A.: From log-d eterminant inequalities to Gauss- ian entanglement via recoverability theory. IEEE Trans. In form. Theory, 63(11), 7553-7568 (2017)

2017
[15]

Lieb, E.H., Ruskai, M.B.: Proof of the strong subadditi vity of quantum mechanichal entropy. J. Mathematical Phys. 14, 1938–1941 (1973)

1973
[16]

Lin, M., Sra, S.: Completely strong superadditivity of generalized matrix functions, arXiv preprint, arXiv:1410.1958 (2014)

2014 arXiv
[17]

Econometrica 58(6), 1255–1277 (1990)

Milgrom, P., Roberts, J.: Rationalizability, Learnin g, and Equilibrium in Games with Strate- gic Complementarities. Econometrica 58(6), 1255–1277 (1990)

1990
[18]

Carpathian J

Niculescu, C.P.: The Abel-Steﬀensen inequality in hig her dimensions. Carpathian J. Math. 35(1), 69 - 78 (2019)

2019
[19]

Niculescu, C.P.: A new look at the Hardy-Littlewood-Po lya inequality of majorization. J. Math. Anal. Appl. 501(2), article 125211 (2021)

2021
[21]

Niculescu, C.P., Sra, S.: Functional inequalities for functions with positive diﬀerences on convex cones. Math. Results 78(6), article 217 (2023)

2023
[22]

Analele ¸ stiint ¸iﬁce Univ

Popoviciu, T.: Sur certaines in´ egalit´ es qui caract´ erisent les fonctions convexes. Analele ¸ stiint ¸iﬁce Univ. ”Al. I. Cuza” Ia¸ si, Sect ¸ia I-a Mat.,11, 155–164 (1965)

1965
[23]

Positivity 18, 257–285 (2014)

Ressel, P.: Higher order monotonic functions of severa l variables. Positivity 18, 257–285 (2014)

2014
[24]

De Gruyter, Berlin (2010)

Schilling, R.L., Song, R., Vondraˇ cek, Z.: Bernstein F unctions, Theory and Applications. De Gruyter, Berlin (2010)

2010
[25]

Acta Math

Scott, A.D., Sokal, A.D.: Complete monotonicity for in verse powers of some combinatorially deﬁned polynomials. Acta Math. 213(3), 323-392 (2014)

2014
[26]

Simon, B.: Trace Ideals and their Applications. 2nd Ed. , Mathematical Surveys and Mono- graphs 120, American Mathematical Society, Providence, R.I. (2005)

2005
[27]

Linear Algebra Appl

Tie, L., Cai, K.-Y., Lin, Y.: Rearrangement inequaliti es for Hermitian matrices. Linear Algebra Appl. 434, 443-456 (2011)

2011
[28]

2nd ed., Springer, New York (2011) Department of Mathematics, University of Craiova, Craiova 2 00585, Romania Email address : constantin.p.niculescu@gmail.com

Zhang, F.: Matrix Theory: Basic Results and Techniques . 2nd ed., Springer, New York (2011) Department of Mathematics, University of Craiova, Craiova 2 00585, Romania Email address : constantin.p.niculescu@gmail.com

2011

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [20]

A Contemporary Approach

Niculescu, C.P., Persson, L.-E.: Convex Functions and Their Applications. A Contemporary Approach. 3rd Ed., CMS Books in Mathematics Vol. 23, Springer-Verlag, New York (2025)

work page 2025

[2] [1]

Graduate Studies in Mathem atics vol

Aliprantis, C.D.[ ?], Tourky, R.: Cones and Duality. Graduate Studies in Mathem atics vol. 84, American Mathematical Society, Providence, R.I. (2007)

work page 2007

[3] [2]

Amann, H.: Multiple positive ﬁxed points of asymptotica lly linear maps. J. Functional Anal- ysis 17(2), 174-213 (1974)

work page 1974

[4] [3]

Acta Mathematica Hun- garica 128(4), 386–394 (2010)

Audenaert, K., Hiai, F., Petz, D.: Strongly subadditive functions. Acta Mathematica Hun- garica 128(4), 386–394 (2010)

work page 2010

[5] [4]

In: International Conferen ce on Engineering Optimization (EngOpt 2008), Rio de Janeiro, June 1–5, 2008, Paper # 0615, P rinted and CDROM Pro- ceedings, ISBN 978-85-7650-152-7

Bouhtou, M., Gaubert, S., Sagnol, G.: Optimization of ne twork traﬃc measurement: a semi- deﬁnite programming approach. In: International Conferen ce on Engineering Optimization (EngOpt 2008), Rio de Janeiro, June 1–5, 2008, Paper # 0615, P rinted and CDROM Pro- ceedings, ISBN 978-85-7650-152-7

work page 2008

[6] [5]

Cambr idge University Press, Cambridge, UK (2004)

Boyd, S., Vandenberghe, L.: Convex Optimization. Cambr idge University Press, Cambridge, UK (2004)

work page 2004

[7] [6]

Contemporary Mathematics 529, 73–140 (2010)

Carlen, E.A.: Trace inequalities and quantum entropy: A n intro- ductory course. Contemporary Mathematics 529, 73–140 (2010). URL http://www.ueltschi.org/AZschool/notes/EricCarlen.pdf

work page 2010

[8] [7]

Choquet, G.: Deux exemples classiques de repr´ esentati on int´ egrale. Enseign. Math. 15(2), 63-75 (1969)

work page 1969