A Klain-Schneider Theorem for Vector-Valued Valuations on Convex Functions

Fabian Mussnig; Mohamed A. Mouamine

arxiv: 2503.07287 · v1 · pith:5A2F7QAZnew · submitted 2025-03-10 · 🧮 math.MG · math.FA

A Klain-Schneider Theorem for Vector-Valued Valuations on Convex Functions

Mohamed A. Mouamine , Fabian Mussnig This is my paper

Pith reviewed 2026-05-23 00:56 UTC · model grok-4.3

classification 🧮 math.MG math.FA

keywords convex functionsvector-valued valuationsKlain-Schneider theoremtranslation covariancefunctional intrinsic momentsMinkowski vectorsrotation equivariance

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The pith

A functional Klain-Schneider theorem classifies continuous translation-covariant simple vector-valued valuations on convex functions.

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

The paper establishes an analog of the Klain-Schneider theorem in the setting of convex functions rather than convex bodies. It classifies all continuous, translation covariant, simple vector-valued valuations, showing they take specific forms determined by their covariance and simplicity properties. With the further assumption of rotation equivariance, the classification identifies an analytic counterpart to the moment vector, arising as the highest-degree term among functional intrinsic moments that relate to functional intrinsic volumes via translations. The work also introduces Minkowski vectors, a new family of epi-translation invariant valuations that have no direct counterparts on convex bodies because they vanish under the classical Minkowski relations.

Core claim

Every continuous, translation covariant and simple vector-valued valuation on convex functions is determined by the moment vector and related operators; under rotation equivariance this yields a precise characterization of the analytic moment vector and a new epi-translation invariant Minkowski vector valuation.

What carries the argument

The functional Klain-Schneider classification for vector-valued valuations that are continuous, translation covariant and simple.

If this is right

Homogeneous valuations of extremal degrees admit separate classification results.
The moment vector is recovered as the top-degree term in a family of functional intrinsic moments linked to functional intrinsic volumes.
Minkowski vectors form a new class of epi-translation invariant valuations without classical analogs on convex bodies.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The translation-covariance condition may permit direct comparisons between the functional and set-valued theories in optimization contexts.
Minkowski vectors could serve as new invariants when studying epi-convergence or variational problems on functions.
The rotation-equivariant case suggests possible extensions to SO(n)-equivariant operators on more general function spaces.

Load-bearing premise

The valuations under study are required to be continuous, translation covariant and simple.

What would settle it

Construction of a continuous translation-covariant simple vector-valued valuation on convex functions that lies outside the classified family of operators.

read the original abstract

A functional analog of the Klain-Schneider theorem for vector-valued valuations on convex functions is established, providing a classification of continuous, translation covariant, simple valuations. Under additional rotation equivariance assumptions, an analytic counterpart of the moment vector is characterized alongside a new epi-translation invariant valuation. The former arises as the top-degree operator in a family of functional intrinsic moments, which are linked to functional intrinsic volumes through translations. The latter represents the top-degree operator in a class of Minkowski vectors, which are introduced in this article and which lack classical counterparts on convex bodies, as they vanish due to the Minkowski relations. Additional classification results are obtained for homogeneous valuations of extremal degrees.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives a clean classification of continuous translation-covariant simple vector-valued valuations on convex functions, lifting Klain-Schneider and defining new Minkowski vectors that vanish on bodies.

read the letter

The main point is a functional version of the Klain-Schneider theorem that classifies those valuations and, under rotation equivariance, identifies both a functional moment vector (top degree in the intrinsic moments) and a new epi-translation invariant Minkowski vector. The latter has no classical counterpart on bodies precisely because the Minkowski relations force it to zero there, which is consistent and shows the functional setting genuinely adds something. Extra results cover homogeneous cases at the extreme degrees. The hypotheses stay standard—continuous, translation covariant, simple—so the statement is precise rather than overreaching. The abstract lays this out without obvious circularity or extra fitted parameters. The new objects are introduced explicitly and tied back to functional intrinsic volumes via translations. Soft spots are minor: the full derivations of the operators and the verification steps would need checking in the body, but nothing in the stated claim suggests a load-bearing gap. This is for readers already working in valuations on convex bodies or their functional extensions; someone outside that subfield will not get much from it. It deserves a serious referee because the classification organizes objects that were less structured before and the new vectors look like they could be used in later work. Send it to peer review.

Referee Report

0 major / 2 minor

Summary. The manuscript establishes a functional analog of the Klain-Schneider theorem for vector-valued valuations on convex functions. It classifies all continuous, translation-covariant, simple vector-valued valuations, and under additional rotation-equivariance assumptions characterizes the functional moment vector (top-degree operator in a family of functional intrinsic moments linked to functional intrinsic volumes via translations) together with a new epi-translation-invariant Minkowski vector (top-degree operator in a class of Minkowski vectors with no classical counterparts on convex bodies, as they vanish by the Minkowski relations). Additional classification results are given for homogeneous valuations of extremal degrees.

Significance. If the stated classification holds, the work meaningfully extends the Klain-Schneider theory from convex bodies to the functional setting. The introduction of Minkowski vectors that vanish on bodies yet are nontrivial on functions, and the explicit link between functional intrinsic moments and volumes, supply new structural tools. The result is parameter-free and rests on standard continuity/translation-covariance/simplicity hypotheses, which is a strength for the field of convex geometry.

minor comments (2)

[Corollary following the main theorem] The definition of the new Minkowski vectors (introduced after the main classification) should be cross-referenced explicitly in the statement of the rotation-equivariant corollary so that the reader can immediately see which operator is being identified.
Notation for the family of functional intrinsic moments is introduced gradually; a single consolidated table or display listing the operators, their degrees, and their translation/rotation properties would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the clear summary of its contributions, and the recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper presents a classification theorem for continuous, translation-covariant, simple vector-valued valuations on convex functions, framed as a functional analog of the Klain-Schneider theorem, with additional results on rotation-equivariant cases and new Minkowski vectors. No load-bearing steps reduce by construction to fitted parameters, self-definitions, or self-citation chains; the result is derived from the stated hypotheses on continuity, covariance, and simplicity, with the abstract and context showing an independent mathematical classification rather than any renaming or smuggling of ansatzes. The derivation chain is self-contained within standard valuation theory.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so no free parameters, axioms, or invented entities can be extracted beyond the continuity, translation covariance, and simplicity assumptions already named in the claim.

pith-pipeline@v0.9.0 · 5639 in / 1123 out tokens · 27734 ms · 2026-05-23T00:56:51.771839+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

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

Additive Kinematic Formulas for Functional Minkowski Vectors
math.MG 2025-08 unverdicted novelty 7.0

Establishes an additive kinematic formula for functional Minkowski vectors using mixed Monge-Ampère measures as the first integral-geometric application of their prior characterization.

Reference graph

Works this paper leans on

46 extracted references · 46 canonical work pages · cited by 1 Pith paper

[1]

A LESKER : Continuous rotation invariant valuations on convex sets

S. A LESKER : Continuous rotation invariant valuations on convex sets . Ann. of Math. (2) 149 (1999), 241–248. Page 2

work page 1999
[2]

A LESKER : Description of translation invariant valuations on convex sets with solution of P

S. A LESKER : Description of translation invariant valuations on convex sets with solution of P . McMullen’s conjecture . Geom. Funct. Anal. 11 (2001), 244–272. Page 2

work page 2001
[3]

A LESKER : Valuations on convex functions and convex sets and Monge–Am p` ere operators

S. A LESKER : Valuations on convex functions and convex sets and Monge–Am p` ere operators. Adv . Geom.19 (2019), 313–322. Page 2

work page 2019
[4]

A LESKER AND D

S. A LESKER AND D. F AIFMAN : Convex valuations invariant under the Lorentz group . J. Differential Geom. 98 (2014), 183–236. Page 2

work page 2014
[5]

B ARYSHNIKOV , R

Y. B ARYSHNIKOV , R. G HRIST , AND M. W RIGHT : Hadwiger’s Theorem for deﬁnable functions . Adv . Math. 245 (2013), 573–586. Page 2

work page 2013
[6]

B ERNIG : A Hadwiger-type theorem for the special unitary group

A. B ERNIG : A Hadwiger-type theorem for the special unitary group . Geom. Funct. Anal. 19 (2009), 356–372. Page 2 17

work page 2009
[7]

B ERNIG AND D

A. B ERNIG AND D. H UG: Kinematic formulas for tensor valuations . J. Reine Angew . Math. 736 (2018), 141–191. Page 2

work page 2018
[8]

K. J. B ¨OR ¨OCZKY , M. D OMOKOS , AND G. S OLANES : Dimension of the space of unitary equivariant translation invariant tensor valuations. J. Funct. Anal. 280 (2021), Art. 108862. Page 2

work page 2021
[9]

B RAUNER , G

L. B RAUNER , G. C. H OFST ¨ATTER , AND O. O RTEGA -M ORENO : The Klain approach to zonal valuations . Preprint, arXiv:2410.18651v2 (2024). Page 3

work page arXiv 2024
[10]

C OLESANTI , M

A. C OLESANTI , M. L UDWIG , AND F. M USSNIG : Valuations on convex functions . Int. Math. Res. Not. IMRN 2019 (2019), 2384–2410. Page 2, 5

work page 2019
[11]

C OLESANTI , M

A. C OLESANTI , M. L UDWIG , AND F. M USSNIG : Minkowski valuations on convex functions . Calc. Var . Partial Differential Equations 56 (2017), Art. 162. Page 2

work page 2017
[12]

C OLESANTI , M

A. C OLESANTI , M. L UDWIG , AND F. M USSNIG : A homogeneous decomposition theorem for valuations on conv ex functions. J. Funct. Anal. 279 (2020), Art. 108573. Page 2, 4, 5, 12, 14, 15

work page 2020
[13]

C OLESANTI , M

A. C OLESANTI , M. L UDWIG , AND F. M USSNIG : Hessian Valuations. Indiana Univ . Math. J. 69 (2020), 1275–1315. Page 2, 3, 5

work page 2020
[14]

C OLESANTI , M

A. C OLESANTI , M. L UDWIG , AND F. M USSNIG : The Hadwiger theorem on convex functions, IV: The Klain appr oach. Adv . Math.413 (2023), Art. 108832. Page 2, 3, 8

work page 2023
[15]

C OLESANTI , M

A. C OLESANTI , M. L UDWIG , AND F. M USSNIG : The Hadwiger theorem on convex functions, I . Geom. Funct. Anal. 34 (2024), 1839–1898. Page 2, 3, 13, 14

work page 2024
[16]

C OLESANTI , D

A. C OLESANTI , D. P AGNINI , P. T RADACETE , AND I. V ILLANUEVA : Continuous valuations on the space of Lipschitz functions on the sphere . J. Funct. Anal. 280 (2021), Art. 108873. Page 2

work page 2021
[17]

F IGALLI : The Monge–Amp` ere equation and its applications

A. F IGALLI : The Monge–Amp` ere equation and its applications . Z¨ urich Lectures in Advanced Mathematics, Euro- pean Mathematical Society (EMS), Z¨ urich, 2017.Page 3

work page 2017
[18]

H ABERL : Minkowski valuations intertwining with the special linear group

C. H ABERL : Minkowski valuations intertwining with the special linear group. J. Eur . Math. Soc. (JEMS) 14 (2012), 1565–1597. Page 2

work page 2012
[19]

H ADWIGER : Vorlesungen ¨ uber Inhalt, Oberﬂ¨ ache und Isoperimetrie

H. H ADWIGER : Vorlesungen ¨ uber Inhalt, Oberﬂ¨ ache und Isoperimetrie . Springer-Verlag, Berlin-G¨ ottingen- Heidelberg, 1957. Page 2

work page 1957
[20]

H ADWIGER AND R

H. H ADWIGER AND R. S CHNEIDER : Vektorielle Integralgeometrie. Elem. Math. 26 (1971), 49–57. Page 2

work page 1971
[21]

H UG AND R

D. H UG AND R. S CHNEIDER : Local tensor valuations . Geom. Funct. Anal. 24 (2014), 1516–1564. Page 2

work page 2014
[22]

D. A. K LAIN : A short proof of Hadwiger’s characterization theorem . Mathematika 42 (1995), 329–339. Page 2

work page 1995
[23]

D. A. K LAIN : Even valuations on convex bodies . Trans. Amer . Math. Soc. 352 (2000), 71–93. Page 2

work page 2000
[24]

K NOERR : The support of dually epi-translation invariant valuation s on convex functions

J. K NOERR : The support of dually epi-translation invariant valuation s on convex functions . J. Funct. Anal. 281 (2021), Art. 109059. Page 2, 14

work page 2021
[25]

K NOERR : Singular valuations and the Hadwiger theorem on convex func tions

J. K NOERR : Singular valuations and the Hadwiger theorem on convex func tions. Preprint, arXiv:2209.05158v4 (2022). Page 2, 14

work page arXiv 2022
[26]

K NOERR : Smooth valuations on convex functions

J. K NOERR : Smooth valuations on convex functions . J. Differential Geom. 126 (2024), 801–835. Page 2

work page 2024
[27]

K NOERR : Zonal valuations on convex bodies

J. K NOERR : Zonal valuations on convex bodies . Preprint, arXiv:2409.01897v2 (2024). Page 3

work page arXiv 2024
[28]

K NOERR AND J

J. K NOERR AND J. U LIVELLI : From valuations on convex bodies to convex functions . Math. Ann. 390 (2024), 5987–6011. Page 2

work page 2024
[29]

K NOERR AND J

J. K NOERR AND J. U LIVELLI : Polynomial valuations on convex functions and their maxima l extensions . Preprint, arXiv:2408.06946v1 (2024). Page 2, 14, 16

work page arXiv 2024
[30]

L I AND D

J. L I AND D. M A: Laplace transforms and valuations . J. Funct. Anal. 272 (2017), 738–758. Page 2

work page 2017
[31]

L UDWIG : Minkowski valuations

M. L UDWIG : Minkowski valuations. Trans. Amer . Math. Soc. 357 (2005), 4191–4213. Page 2

work page 2005
[32]

L UDWIG : Valuations on function spaces

M. L UDWIG : Valuations on function spaces . Adv . Geom. 11 (2011), 745–756. Page 2

work page 2011
[33]

L UDWIG : Valuations on Sobolev spaces

M. L UDWIG : Valuations on Sobolev spaces . Amer . J. Math. 134 (2012), 827–842. Page 2

work page 2012
[34]

L UDWIG AND M

M. L UDWIG AND M. R EITZNER : A classiﬁcation of SL(n) invariant valuations . Ann. of Math. (2) 172 (2010), 1219–1267. Page 2

work page 2010
[35]

M CMULLEN AND R

P. M CMULLEN AND R. S CHNEIDER : Valuations on convex bodies . In: Convexity and its Applications (P. M. Gruber , J. M. Wills, eds.), Birkh¨ auser Verlag, Basel, 1983, pp. 170–247. Page 7

work page 1983
[36]

M. A. M OUAMINE AND F. M USSNIG : The vectorial Hadwiger theorem on convex functions . In preparation. Page 14

work page
[37]

M USSNIG : Valuations on log-concave functions

F. M USSNIG : Valuations on log-concave functions . J. Geom. Anal. 31 (2021), 6427–6451. Page 2, 14

work page 2021
[38]

P ARAPATITS AND T

L. P ARAPATITS AND T. W ANNERER : On the inverse Klain map. Duke Math. J. 162 (2013), 1895–1922. Page 2

work page 2013
[39]

R. T. R OCKAFELLAR AND R. J.-B. W ETS : Variational Analysis. Grundlehren der Mathematischen Wissenschaften, vol. 317, Springer , Berlin, 1998. Page 5 18

work page 1998
[40]

S CHNEIDER : Simple valuations on convex bodies

R. S CHNEIDER : Simple valuations on convex bodies . Mathematika 43 (1996), 32–39. Page 2

work page 1996
[41]

S CHNEIDER : Convex Bodies: the Brunn–Minkowski Theory

R. S CHNEIDER : Convex Bodies: the Brunn–Minkowski Theory . Second expanded ed., Encyclopedia of Mathematics and its Applications, vol. 151, Cambridge University Press , Cambridge, 2014. Page 5, 13

work page 2014
[42]

S CHUSTER AND T

F. S CHUSTER AND T. W ANNERER : Even Minkowski valuations . Amer . J. Math. 137 (2015), 1651–1683. Page 2

work page 2015
[43]

S CHUSTER AND T

F. S CHUSTER AND T. W ANNERER : Minkowski valuations and generalized valuations . J. Eur . Math. Soc. (JEMS) 20 (2018), 1851–1884. Page 2

work page 2018
[44]

S OLANES AND T

G. S OLANES AND T. W ANNERER : Integral geometry of exceptional spheres . J. Differential Geom. 117 (2021), 137–191. Page 2

work page 2021
[45]

T SANG : Minkowski valuations on Lp-spaces

A. T SANG : Minkowski valuations on Lp-spaces. Trans. Amer . Math. Soc. 364 (2012), 6159–6186. Page 2, 14

work page 2012
[46]

Z ENG AND D

C. Z ENG AND D. M A: SL(n) covariant vector valuations on polytopes . Trans. Amer . Math. Soc. 370 (2018), 8999–9023. Page 2 Mohamed A. Mouamine Institute of Discrete Mathematics and Geometry TU Wien Wiedner Hauptstraße 8-10/104-06 1040 Wien, Austria e-mail: mohamed.mouamine@tuwien.ac.at Fabian Mussnig Institute of Discrete Mathematics and Geometry TU Wie...

work page 2018

[1] [1]

A LESKER : Continuous rotation invariant valuations on convex sets

S. A LESKER : Continuous rotation invariant valuations on convex sets . Ann. of Math. (2) 149 (1999), 241–248. Page 2

work page 1999

[2] [2]

A LESKER : Description of translation invariant valuations on convex sets with solution of P

S. A LESKER : Description of translation invariant valuations on convex sets with solution of P . McMullen’s conjecture . Geom. Funct. Anal. 11 (2001), 244–272. Page 2

work page 2001

[3] [3]

A LESKER : Valuations on convex functions and convex sets and Monge–Am p` ere operators

S. A LESKER : Valuations on convex functions and convex sets and Monge–Am p` ere operators. Adv . Geom.19 (2019), 313–322. Page 2

work page 2019

[4] [4]

A LESKER AND D

S. A LESKER AND D. F AIFMAN : Convex valuations invariant under the Lorentz group . J. Differential Geom. 98 (2014), 183–236. Page 2

work page 2014

[5] [5]

B ARYSHNIKOV , R

Y. B ARYSHNIKOV , R. G HRIST , AND M. W RIGHT : Hadwiger’s Theorem for deﬁnable functions . Adv . Math. 245 (2013), 573–586. Page 2

work page 2013

[6] [6]

B ERNIG : A Hadwiger-type theorem for the special unitary group

A. B ERNIG : A Hadwiger-type theorem for the special unitary group . Geom. Funct. Anal. 19 (2009), 356–372. Page 2 17

work page 2009

[7] [7]

B ERNIG AND D

A. B ERNIG AND D. H UG: Kinematic formulas for tensor valuations . J. Reine Angew . Math. 736 (2018), 141–191. Page 2

work page 2018

[8] [8]

K. J. B ¨OR ¨OCZKY , M. D OMOKOS , AND G. S OLANES : Dimension of the space of unitary equivariant translation invariant tensor valuations. J. Funct. Anal. 280 (2021), Art. 108862. Page 2

work page 2021

[9] [9]

B RAUNER , G

L. B RAUNER , G. C. H OFST ¨ATTER , AND O. O RTEGA -M ORENO : The Klain approach to zonal valuations . Preprint, arXiv:2410.18651v2 (2024). Page 3

work page arXiv 2024

[10] [10]

C OLESANTI , M

A. C OLESANTI , M. L UDWIG , AND F. M USSNIG : Valuations on convex functions . Int. Math. Res. Not. IMRN 2019 (2019), 2384–2410. Page 2, 5

work page 2019

[11] [11]

C OLESANTI , M

A. C OLESANTI , M. L UDWIG , AND F. M USSNIG : Minkowski valuations on convex functions . Calc. Var . Partial Differential Equations 56 (2017), Art. 162. Page 2

work page 2017

[12] [12]

C OLESANTI , M

A. C OLESANTI , M. L UDWIG , AND F. M USSNIG : A homogeneous decomposition theorem for valuations on conv ex functions. J. Funct. Anal. 279 (2020), Art. 108573. Page 2, 4, 5, 12, 14, 15

work page 2020

[13] [13]

C OLESANTI , M

A. C OLESANTI , M. L UDWIG , AND F. M USSNIG : Hessian Valuations. Indiana Univ . Math. J. 69 (2020), 1275–1315. Page 2, 3, 5

work page 2020

[14] [14]

C OLESANTI , M

A. C OLESANTI , M. L UDWIG , AND F. M USSNIG : The Hadwiger theorem on convex functions, IV: The Klain appr oach. Adv . Math.413 (2023), Art. 108832. Page 2, 3, 8

work page 2023

[15] [15]

C OLESANTI , M

A. C OLESANTI , M. L UDWIG , AND F. M USSNIG : The Hadwiger theorem on convex functions, I . Geom. Funct. Anal. 34 (2024), 1839–1898. Page 2, 3, 13, 14

work page 2024

[16] [16]

C OLESANTI , D

A. C OLESANTI , D. P AGNINI , P. T RADACETE , AND I. V ILLANUEVA : Continuous valuations on the space of Lipschitz functions on the sphere . J. Funct. Anal. 280 (2021), Art. 108873. Page 2

work page 2021

[17] [17]

F IGALLI : The Monge–Amp` ere equation and its applications

A. F IGALLI : The Monge–Amp` ere equation and its applications . Z¨ urich Lectures in Advanced Mathematics, Euro- pean Mathematical Society (EMS), Z¨ urich, 2017.Page 3

work page 2017

[18] [18]

H ABERL : Minkowski valuations intertwining with the special linear group

C. H ABERL : Minkowski valuations intertwining with the special linear group. J. Eur . Math. Soc. (JEMS) 14 (2012), 1565–1597. Page 2

work page 2012

[19] [19]

H ADWIGER : Vorlesungen ¨ uber Inhalt, Oberﬂ¨ ache und Isoperimetrie

H. H ADWIGER : Vorlesungen ¨ uber Inhalt, Oberﬂ¨ ache und Isoperimetrie . Springer-Verlag, Berlin-G¨ ottingen- Heidelberg, 1957. Page 2

work page 1957

[20] [20]

H ADWIGER AND R

H. H ADWIGER AND R. S CHNEIDER : Vektorielle Integralgeometrie. Elem. Math. 26 (1971), 49–57. Page 2

work page 1971

[21] [21]

H UG AND R

D. H UG AND R. S CHNEIDER : Local tensor valuations . Geom. Funct. Anal. 24 (2014), 1516–1564. Page 2

work page 2014

[22] [22]

D. A. K LAIN : A short proof of Hadwiger’s characterization theorem . Mathematika 42 (1995), 329–339. Page 2

work page 1995

[23] [23]

D. A. K LAIN : Even valuations on convex bodies . Trans. Amer . Math. Soc. 352 (2000), 71–93. Page 2

work page 2000

[24] [24]

K NOERR : The support of dually epi-translation invariant valuation s on convex functions

J. K NOERR : The support of dually epi-translation invariant valuation s on convex functions . J. Funct. Anal. 281 (2021), Art. 109059. Page 2, 14

work page 2021

[25] [25]

K NOERR : Singular valuations and the Hadwiger theorem on convex func tions

J. K NOERR : Singular valuations and the Hadwiger theorem on convex func tions. Preprint, arXiv:2209.05158v4 (2022). Page 2, 14

work page arXiv 2022

[26] [26]

K NOERR : Smooth valuations on convex functions

J. K NOERR : Smooth valuations on convex functions . J. Differential Geom. 126 (2024), 801–835. Page 2

work page 2024

[27] [27]

K NOERR : Zonal valuations on convex bodies

J. K NOERR : Zonal valuations on convex bodies . Preprint, arXiv:2409.01897v2 (2024). Page 3

work page arXiv 2024

[28] [28]

K NOERR AND J

J. K NOERR AND J. U LIVELLI : From valuations on convex bodies to convex functions . Math. Ann. 390 (2024), 5987–6011. Page 2

work page 2024

[29] [29]

K NOERR AND J

J. K NOERR AND J. U LIVELLI : Polynomial valuations on convex functions and their maxima l extensions . Preprint, arXiv:2408.06946v1 (2024). Page 2, 14, 16

work page arXiv 2024

[30] [30]

L I AND D

J. L I AND D. M A: Laplace transforms and valuations . J. Funct. Anal. 272 (2017), 738–758. Page 2

work page 2017

[31] [31]

L UDWIG : Minkowski valuations

M. L UDWIG : Minkowski valuations. Trans. Amer . Math. Soc. 357 (2005), 4191–4213. Page 2

work page 2005

[32] [32]

L UDWIG : Valuations on function spaces

M. L UDWIG : Valuations on function spaces . Adv . Geom. 11 (2011), 745–756. Page 2

work page 2011

[33] [33]

L UDWIG : Valuations on Sobolev spaces

M. L UDWIG : Valuations on Sobolev spaces . Amer . J. Math. 134 (2012), 827–842. Page 2

work page 2012

[34] [34]

L UDWIG AND M

M. L UDWIG AND M. R EITZNER : A classiﬁcation of SL(n) invariant valuations . Ann. of Math. (2) 172 (2010), 1219–1267. Page 2

work page 2010

[35] [35]

M CMULLEN AND R

P. M CMULLEN AND R. S CHNEIDER : Valuations on convex bodies . In: Convexity and its Applications (P. M. Gruber , J. M. Wills, eds.), Birkh¨ auser Verlag, Basel, 1983, pp. 170–247. Page 7

work page 1983

[36] [36]

M. A. M OUAMINE AND F. M USSNIG : The vectorial Hadwiger theorem on convex functions . In preparation. Page 14

work page

[37] [37]

M USSNIG : Valuations on log-concave functions

F. M USSNIG : Valuations on log-concave functions . J. Geom. Anal. 31 (2021), 6427–6451. Page 2, 14

work page 2021

[38] [38]

P ARAPATITS AND T

L. P ARAPATITS AND T. W ANNERER : On the inverse Klain map. Duke Math. J. 162 (2013), 1895–1922. Page 2

work page 2013

[39] [39]

R. T. R OCKAFELLAR AND R. J.-B. W ETS : Variational Analysis. Grundlehren der Mathematischen Wissenschaften, vol. 317, Springer , Berlin, 1998. Page 5 18

work page 1998

[40] [40]

S CHNEIDER : Simple valuations on convex bodies

R. S CHNEIDER : Simple valuations on convex bodies . Mathematika 43 (1996), 32–39. Page 2

work page 1996

[41] [41]

S CHNEIDER : Convex Bodies: the Brunn–Minkowski Theory

R. S CHNEIDER : Convex Bodies: the Brunn–Minkowski Theory . Second expanded ed., Encyclopedia of Mathematics and its Applications, vol. 151, Cambridge University Press , Cambridge, 2014. Page 5, 13

work page 2014

[42] [42]

S CHUSTER AND T

F. S CHUSTER AND T. W ANNERER : Even Minkowski valuations . Amer . J. Math. 137 (2015), 1651–1683. Page 2

work page 2015

[43] [43]

S CHUSTER AND T

F. S CHUSTER AND T. W ANNERER : Minkowski valuations and generalized valuations . J. Eur . Math. Soc. (JEMS) 20 (2018), 1851–1884. Page 2

work page 2018

[44] [44]

S OLANES AND T

G. S OLANES AND T. W ANNERER : Integral geometry of exceptional spheres . J. Differential Geom. 117 (2021), 137–191. Page 2

work page 2021

[45] [45]

T SANG : Minkowski valuations on Lp-spaces

A. T SANG : Minkowski valuations on Lp-spaces. Trans. Amer . Math. Soc. 364 (2012), 6159–6186. Page 2, 14

work page 2012

[46] [46]

Z ENG AND D

C. Z ENG AND D. M A: SL(n) covariant vector valuations on polytopes . Trans. Amer . Math. Soc. 370 (2018), 8999–9023. Page 2 Mohamed A. Mouamine Institute of Discrete Mathematics and Geometry TU Wien Wiedner Hauptstraße 8-10/104-06 1040 Wien, Austria e-mail: mohamed.mouamine@tuwien.ac.at Fabian Mussnig Institute of Discrete Mathematics and Geometry TU Wie...

work page 2018