A homogeneous decomposition theorem for valuations on convex functions

A. Colesanti; F. Mussnig; M. Ludwig

REVIEW 3 major objections 3 minor 39 references

A homogeneous decomposition theorem for valuations on convex functions

T0 review · 3 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper proves that every continuous valuation on super-coercive convex functions that is invariant under translations of the epigraph splits into a sum of valuations homogeneous of degrees 0 through n, giving a functional analogue of…

desk verdict Genuinely new structural theory for valuations on convex functions; main theorems are probably right, but the proof of Lemma 16 as written has a false step and Theorem 14 is stated without proof. read the letter →

arxiv 1908.10724 v1 pith:BDQUMYUC submitted 2019-08-28 math.MG math.FA

A. Colesanti , M. Ludwig , F. Mussnig This is my paper · ORCID

classification math.MGmath.FA MSC 52B4526B2552A2152A41

keywords convexfunctionsvaluationshomogeneousdecompositionepi-translationinvariancesuper-coerciveHessianmeasuresLegendretransforminf-convolution

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

Valuations are additive functionals that generalize measures, and for convex bodies McMullen's theorem says every continuous translation-invariant valuation splits into homogeneous pieces. This paper proves the functional version: every continuous valuation on super-coercive convex functions that is epi-translation invariant—unchanged when the epigraph is translated horizontally or shifted vertically—is the sum of valuations homogeneous of degrees $0$ through $n$. The decomposition matters because it gives a functional analogue of Minkowski mixed volumes, with inf-convolution playing the role of addition. The paper also classifies the top-degree component as an integral of a compactly supported function of the gradient, and via the Legendre transform obtains the same structure for finite-valued convex functions.

What carries the argument

The central object is epi-multiplication, $(\lambda u)(x)=\lambda u(x/\lambda)$, which rescales the epigraph of $u$ by $\lambda$, together with the test functions $\ell_y+I_K$ that embed convex-body valuations into the function space. Epi-translation invariance is invariance under horizontal translations and vertical constants. The load-bearing mechanism is the induced family of valuations $K\mapsto Z(\ell_y+I_K)$ on convex bodies: the classical linear-action decomposition gives components indexed by degree, a Vandermonde inversion extracts them, and Lemma 16 lifts the resulting identity from these test functions to all super-coercive convex functions. The inclusion-exclusion principle behind Lemma 16 rests on an extension result for valuations on the lattice of finite minima, stated without proof in Section 3. Classification of the top-degree component uses the fact that its induced body valuation is $n$-homogeneous and hence a multiple of volume, which forces the integral representation involving $\zeta(\nabla u)$.

What would settle it

Construct a continuous valuation on $\operatorname{Conv}(\mathbb{R}^n)$ that admits two different continuous extensions to the lattice of finite minima, or exhibit a continuous epi-translation invariant valuation $Z$ on $\operatorname{Conv}_{sc}(\mathbb{R}^n)$ for which the Vandermonde-inverted functionals $Z_i$ fail the valuation property on piecewise affine functions; either observation would refute Theorem 1. A direct numerical check on a one- or two-dimensional example would be a practical place to look.

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Extended reading notes

Core claim

The central claim is Theorem 1: if $Z:\operatorname{Conv}_{sc}(\mathbb{R}^n)\to\mathbb{R}$ is a continuous and epi-translation invariant valuation, then there exist continuous epi-translation invariant valuations $Z_0,\dots,Z_n$ such that $Z_i(\lambda u)=\lambda^i Z_i(u)$ for every $\lambda>0$ and $Z=Z_0+\cdots+Z_n$. The proof evaluates $Z$ on functions $\ell_y+I_K$ built from linear functions and convex-body indicators, where the valuation becomes a translation-invariant valuation on convex bodies, applies the classical homogeneous decomposition, and inverts a Vandermonde system to define the components $Z_i$ before lifting the identity to all of $\operatorname{Conv}_{sc}$ via an inclusion-exclusion principle. A companion result characterizes the degree-$n$ component as $\int_{\operatorname{dom}(u)}\zeta(\nabla u(x))\,dx$ with $\zeta\in C_c(\mathbb{R}^n)$, and degree-$0$ components are constants. By conjugacy these statements transfer to continuous dually epi-translation invariant valuations on finite-valued convex functions.

Load-bearing premise

The load-bearing premise is the unproved extension result stated as Theorem 14: a continuous valuation on $\operatorname{Conv}(\mathbb{R}^n)$ extends uniquely to a valuation on the lattice of finite minima, and if this extension fails, the inclusion-exclusion principle and the lifting lemma that complete the decomposition collapse.

Editorial extensions

If this is right

Every continuous epi-translation invariant valuation on $\operatorname{Conv}_{sc}(\mathbb{R}^n)$ admits a decomposition $Z=\sum_{i=0}^n Z_i$ into valuations of homogeneous degree $i$.
Evaluations of such valuations along inf-convolutions $Z(\lambda_1 u_1\,\square\,\cdots\,\square\,\lambda_k u_k)$ are polynomials in the coefficients, yielding a functional analogue of Minkowski's mixed volume theorem.
The degree-$n$ component is exactly $\int_{\operatorname{dom}(u)}\zeta(\nabla u(x))\,dx$, so the only top-degree invariant information is carried by a compactly supported test function applied to the gradient.
In dimension one every such valuation has the explicit form $\zeta_0+\int_{\operatorname{dom}(u)}\zeta_1(u'(x))\,dx$.
On the larger space of coercive convex functions the same hypotheses force the valuation to be constant, so super-coercivity is the right setting for nontrivial decompositions.

Reading between the lines

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

Editorial inference: because the proof pins the decomposition to values $Z(k u)$ for $k=0,\dots,n$, the same Vandermonde construction should produce homogeneous components for any valuation that is merely continuous and translation invariant on a space where test functions $\ell_y+I_K$ are dense, provided an inclusion-exclusion principle holds.
Editorial inference: the failure without vertical translation invariance, shown by valuations with exponential growth under epi-scaling, suggests that the dichotomy polynomial-versus-exponential growth marks exactly where McMullen-style decomposition is possible; testing other function lattices for the same dichotomy would clarify the boundary.
Editorial inference: a complete proof of the extension result in Section 3 would remove the only unverified step, and a counterexample would not necessarily destroy Theorem 1 if the inclusion-exclusion principle could be proved directly; isolating that step is the most efficient test of the paper's foundation.

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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

Read this for the theory, not for the proof as written. The paper introduces epi-translation invariance and proves a McMullen-type homogeneous decomposition for continuous epi-translation invariant valuations on super-coercive convex functions, plus a classification of the n-homogeneous ones. That is a real step forward: there was no translation-invariant theory for valuations on convex functions before, and the examples via Hessian valuations show the class is nonempty and nontrivial. The duality to finite-valued convex functions and the polynomiality consequences are natural and useful. I believe the main results are true. The citation pattern is unremarkable; the heavy reliance on the authors' earlier Hessian valuation paper is justified because that is where the machinery comes from.

Now the soft spots. Two are in the proof scaffolding, and one is serious.

First, Theorem 14—the unique extension of a continuous valuation from Conv to the lattice of finite minima—is stated with the proof omitted, called a slight modification of Groemer. It is load-bearing: Theorem 13 and hence Lemma 16 depend on it. Even if it is standard, a paper this ambitious should include the modification, or at least a precise reference to where it appears.

Second, Lemma 16's proof is not just incomplete; it has a false step. For u = ⋀_i(w_i+I_P_i), after inclusion-exclusion you need Z(⋁_{j∈J}(w_j+I_P_j))=0. The paper says this follows from (5). It does not. That maximum is finite on ⋂_{j∈J}P_j and equals max w_j there, which need not be affine. The square-cut-by-diagonal example in the stress-test is exactly right: on the diagonal segment, max(x+y, -x-y)=2|t|. So the sentence 'This follows from (5)' is wrong. You can probably fix this by induction on the dimension of the intersection polytope, using a polytopal subdivision, but the argument needs to be written. Since Lemma 16 is used in Theorems 1, 25 and 30, the proof of the main theorem has a gap at a load-bearing point.

There is also a small sign issue in Proposition 27 (the cylinder height needs |ζ(y_k)| or a fixed-sign subsequence), but that is cosmetic.

Net: the architecture of the proof is sensible and the gaps look repairable rather than fatal. But as submitted, the central lemma is not proven. I would send it to a serious referee, not desk reject. The referee should require a complete proof of Theorem 14 and a correct proof of Lemma 16. If those come back, this is a paper I would be happy to see published.

Referee Report

3 major / 3 minor

Summary. The paper establishes a homogeneous decomposition theorem for continuous and epi-translation invariant valuations on the space of super-coercive convex functions, in analogy with McMullen's theorem for convex bodies. It proves that every such valuation is the sum of continuous, epi-translation invariant valuations that are epi-homogeneous of degrees 0 through n, classifies the degree-n valuations via an integral representation with a compactly supported density of the gradient, and transfers these results by Legendre duality to valuations on finite-valued convex functions. The proofs use McMullen's homogeneous decomposition for induced valuations on convex bodies, a Vandermonde inversion, the inclusion-exclusion principle for valuations on convex functions, and Hadwiger's classification theorem for n-homogeneous translation invariant valuations on convex bodies. The paper also gives counterexamples showing that vertical translation invariance is necessary and that the corresponding valuations on the larger space of coercive convex functions are constant.

Significance. If the main results are correct, this is a substantial contribution to the functional valuation theory: it provides a genuine functional analogue of McMullen's decomposition theorem and, in the top-degree case, a classification by a compactly supported density, together with a complete one-dimensional description. The proof strategy is largely non-circular and rests on established results for valuations on convex bodies; the Vandermonde argument is clean and the use of Hadwiger's theorem in the only-if direction of the top-degree classification is appropriate. The paper also connects the new classes to Alesker's Hessian valuations and gives explicit examples showing the necessity of the hypotheses. The main concerns are local but load-bearing: an unproved extension theorem and a gap in the proof of a key lemma.

major comments (3)

[Section 3, Lemma 16] The proof step 'This follows from (5)' is not valid, because the pointwise maximum of finitely many truncated affine functions need not be of the form w + I_P for an affine function w. For example, in R^2 let P_1 and P_2 be the two triangles of the square [-1,1]^2 cut by the diagonal y=x, and set w_1=x+y and w_2=-(x+y); then the maximum on P_1∩P_2 restricts to the function 2|t| along the diagonal, which is not affine. Since Lemma 16 is used in the proofs of Theorems 1, 25, and 30, this gap is load-bearing; an induction on the dimension of the domain of the maximum is needed, but no such argument is supplied.
[Section 3, Theorems 14 and 15] The extension theorem for continuous valuations on Conv(R^n) to the lattice of finite minima is stated as a 'slight modification' of Groemer's extension theorem with the proof omitted. This extension is the basis of the inclusion-exclusion principle (Theorem 13), which is used in Lemma 16 and hence in the main arguments of the paper. The authors should either provide a full proof or give a precise reference to a theorem that explicitly covers the lattice of epigraphs of convex functions; as written, the central derivation depends on an unproved statement.
[Section 7, Proposition 27] In the construction of the cylinder C_k, the height is given by 1/ζ(y_k), but ζ(y_k) may be negative, so the set as written is not a cylinder of positive height and the displayed volume formula can be negative. The argument should use |ζ(y_k)| or should first pass to a subsequence along which ζ(y_k) has a fixed sign; with this repair, the compact-support conclusion follows.

minor comments (3)

[Section 7, Eq. (12)] The notation ~Z(K) in equation (12) should be ~Z_y(K), since the valuation defined there depends on y; the subscript appears elsewhere in the proof and its omission here is confusing.
[Section 4, proof of Proposition 19] The proof contains a duplicated word: 'continuous valuation on on Conv(R^n;R)' should be 'continuous valuation on Conv(R^n;R)'.
[Throughout] There are several typographical errors, such as 'epi-translation invariant valuati on' in Theorem 26 and 'F or' at the beginnings of paragraphs; these should be corrected before publication.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 1 is proved via McMullen's theorem and Vandermonde inversion, and the target decomposition is not assumed in the proof.

full rationale

The central claim, Theorem 1, is derived by first restricting the valuation to functions of the form ℓ_y + I_K on convex bodies K, applying McMullen's homogeneous decomposition theorem for translation-invariant valuations on convex bodies, and then using a Vandermonde inversion to construct components Z_i that agree with the homogeneous pieces on these test functions. Lemma 16 is then invoked to lift the identity and homogeneity from test functions to all of Conv_sc(R^n). None of these steps assumes the conclusion; the decomposition is genuinely obtained from an external theorem plus a linear-algebra inversion. The classification results similarly use Hadwiger's theorem for convex bodies and Hessian measure constructions; the 'if' direction of Theorem 2 relies on Proposition 20, which cites the authors' prior Hessian valuation paper [16], but that is an independent published construction and is not identical to the target classification. The paper does contain an omitted proof of Theorem 14 (the extension theorem cited as a 'slight modification' of Groemer's theorem), and the proof of Lemma 16 contains a step that is at least incomplete: the maximum of finitely many truncated affine functions need not itself be of the form w + I_P for an affine w, so the sentence 'This follows from (5)' is not justified as written. These are correctness concerns about missing or defective arguments, not circularity: there is no place where a predicted quantity is reintroduced as an input, where a parameter is fitted and then called a prediction, or where a uniqueness claim is imported solely from the authors' own prior work. No equation in the paper is shown to be equal to its own input by construction. Therefore the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no fitted parameters or new physical entities. The homogeneous decomposition is derived from classical theorems on convex bodies (McMullen, Hadwiger) and a lattice extension theorem (Groemer), plus the authors' previous Hessian valuation theory. All listed axioms are background results used in the proofs, not the result being proved.

assumptions (5)

domain assumption Groemer's extension theorem extends to the lattice of finite minima of convex functions (Theorem 14/15).
Stated without proof in Section 3 as a 'slight modification' of [18]; used to prove the inclusion-exclusion principle (Theorem 13) and Lemma 16.
standard math McMullen's homogeneous decomposition theorem for continuous translation invariant valuations on convex bodies (Theorem 7).
External classical result used in Section 5 to decompose the induced valuations on convex bodies.
standard math Hadwiger's classification of n-homogeneous translation invariant valuations on convex bodies (Theorem 8).
External classical result used in Section 7 to identify the degree-n induced valuation as a multiple of volume.
domain assumption Hessian valuations from the authors' prior work are continuous valuations (Theorem 18, cited from [16]).
Self-cited result used in Theorem 17 and Proposition 19 to construct the valuations appearing in the 'if' part of Theorem 2 and in Section 8. Not reproduced here.
domain assumption Epi-convergence for coercive functions is equivalent to Hausdorff convergence of sublevel sets away from the minimum (Lemma 10).
Follows from [15] and [5]; used to prove continuity of the induced body valuations and the epi-convergence arguments in Sections 5, 7 and 9.

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Pith. "Pith review of A homogeneous decomposition theorem for valuations on convex functions." pith.science (2026). https://pith.science/paper/BDQUMYUC

@misc{pith2026190810724,
  author       = {Pith},
  title        = {Pith review of: A homogeneous decomposition theorem for valuations on convex functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BDQUMYUC}},
  note         = {Machine review of arXiv:1908.10724}
}

abstract

The existence of a homogeneous decomposition for continuous and epi-translation invariant valuations on super-coercive functions is established. Continuous and epi-translation invariant valuations that are epi-homogeneous of degree $n$ are classified. By duality, corresponding results are obtained for valuations on finite-valued convex functions.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Alesker, Continuous rotation invariant valuations on convex sets , Ann

S. Alesker, Continuous rotation invariant valuations on convex sets , Ann. of Math. (2) 149 (1999), 977–1005

work page 1999

[2] [2]

Alesker, Description of translation invariant valuations on convex sets with solution of P

S. Alesker, Description of translation invariant valuations on convex sets with solution of P. McMullen’s conjecture , Geom. Funct. Anal. 11 (2001), 244–272

work page 2001

[3] [3]

Alesker, V aluations on convex functions and convex sets and Monge–Am p`ere operators , Adv

S. Alesker, V aluations on convex functions and convex sets and Monge–Am p`ere operators , Adv. Geom. 19 (2019), 313–322

work page 2019

[4] [4]

Baryshnikov, R

Y . Baryshnikov, R. Ghrist, and M. Wright, Hadwiger’s Theorem for deﬁnable functions , Adv. Math. 245 (2013), 573– 586

work page 2013

[5] [5]

G. Beer, R. T. Rockafellar and R. J.-B. Wets, A characterization of epi-convergence in terms of converge nce of level sets, Proc. Amer. Math. Soc. 116 (1992), 753–761

work page 1992

[6] [6]

Bernig and J

A. Bernig and J. H. G. Fu, Hermitian integral geometry, Ann. of Math. (2) 173 (2011), 907–945

work page 2011

[7] [7]

Bernig, J

A. Bernig, J. H. G. Fu, and G. Solanes, Integral geometry of complex space forms , Geom. Funct. Anal. 24 (2014), 403–492

work page 2014

[8] [8]

Cavallina and A

L. Cavallina and A. Colesanti, Monotone valuations on the space of convex functions , Anal. Geom. Metr. Spaces 3 (2015), 167–211

work page 2015