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arxiv: 2508.07984 · v2 · pith:ZG6WZWBYnew · submitted 2025-08-11 · 🧮 math.MG · math.FA

Additive Kinematic Formulas for Functional Minkowski Vectors

Mohamed A. Mouamine This is my paper

Pith reviewed 2026-05-21 23:08 UTC · model grok-4.3

classification 🧮 math.MG math.FA
keywords functional Minkowski vectorsadditive kinematic formulamixed Monge-Ampère measuresconvex functionsvector-valued valuationsintegral geometrykinematic formulas
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The pith

Functional Minkowski vectors satisfy an additive kinematic formula derived from mixed Monge-Ampère measures.

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

The paper sets out to prove an additive kinematic formula for functional Minkowski vectors defined on convex functions. These vectors form a family of vector-valued valuations whose properties were characterized in earlier work. The proof uses mixed Monge-Ampère measures to integrate over rigid motions and obtain the additive identity. A reader would care because the result supplies the first integral-geometric application of that characterization, lifting classical kinematic techniques to a functional setting.

Core claim

The authors establish an additive kinematic formula for the functional Minkowski vectors using mixed Monge-Ampère measures. These vectors are a natural family of vector-valued valuations on the space of convex functions, as characterized in prior joint work. The formula is obtained by exploiting the additivity and continuity of the mixed measures, marking the first integral-geometric use of the characterization.

What carries the argument

Mixed Monge-Ampère measures, which supply the additive terms that convert the valuation properties of the functional Minkowski vectors into the desired kinematic identity.

If this is right

  • The kinematic integral of any functional Minkowski vector can be rewritten as an additive combination of mixed Monge-Ampère measures.
  • The same additivity extends to averages over the group of rigid motions acting on convex functions.
  • The result confirms that the prior characterization of the vectors is compatible with classical integral-geometric operations.

Where Pith is reading between the lines

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

  • The same mixed-measure technique may produce kinematic formulas for other vector-valued or scalar functional valuations.
  • The approach could be tested numerically on low-dimensional convex functions such as quadratics to check numerical stability.
  • Connections may exist to problems in optimal transport or stochastic geometry where Monge-Ampère-type measures already appear.

Load-bearing premise

The functional Minkowski vectors must behave exactly according to their prior characterization, and the mixed Monge-Ampère measures must satisfy the additivity and continuity properties needed to produce the kinematic identity.

What would settle it

A concrete counter-example consisting of two explicit convex functions for which the integral over the group of rigid motions fails to equal the sum predicted by the mixed Monge-Ampère expression.

read the original abstract

We establish an additive kinematic formula for the functional Minkowski vectors using mixed Monge-Amp\`ere measures. These vectors, recently introduced and characterized by the author and F. Mussnig, form a natural family of vector-valued valuations on the space of convex functions. This result represents the first integral geometric application of this characterization.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript establishes an additive kinematic formula for the functional Minkowski vectors on the space of convex functions. These vector-valued valuations were recently introduced and characterized by the author jointly with F. Mussnig; the present work applies that characterization to derive the first integral-geometric identity in this setting, expressed via mixed Monge-Ampère measures and integration over the rigid-motion group.

Significance. If the derivation holds, the result supplies a natural functional extension of classical kinematic formulas, linking valuation theory for convex bodies with the analysis of convex functions via Monge-Ampère measures. This could enable further integral-geometric applications in the functional setting and provide a template for vector-valued kinematic identities beyond the scalar case.

major comments (2)
  1. [Introduction and statement of the main theorem] The central kinematic identity is obtained by invoking additivity and continuity of the mixed Monge-Ampère measures directly from the prior characterization in the joint work with Mussnig. No independent verification or low-dimensional sanity check (e.g., for strictly convex quadratic functions or indicator functions of polytopes) is supplied in the present manuscript to confirm that these properties survive the integration against the motion-invariant measure.
  2. [Proof of the kinematic formula] For lower-semicontinuous convex functions that are not strictly convex, the Monge-Ampère measures may fail to be additive under Minkowski addition in the precise sense required to interchange the integral over the motion group with the vector-valued valuation. The manuscript should either add an explicit continuity or approximation argument or restrict the domain to a subclass where the interchange is justified.
minor comments (2)
  1. [Preliminaries] Notation for the functional Minkowski vectors and the mixed Monge-Ampère measures should be introduced with a short reminder of the relevant properties from the prior paper to improve readability.
  2. [Introduction] The abstract states that the result is the 'first integral geometric application'; a brief comparison with existing functional kinematic formulas (if any) would clarify the precise novelty.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive suggestions. We respond to each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Introduction and statement of the main theorem] The central kinematic identity is obtained by invoking additivity and continuity of the mixed Monge-Ampère measures directly from the prior characterization in the joint work with Mussnig. No independent verification or low-dimensional sanity check (e.g., for strictly convex quadratic functions or indicator functions of polytopes) is supplied in the present manuscript to confirm that these properties survive the integration against the motion-invariant measure.

    Authors: The additivity and continuity of the mixed Monge-Ampère measures are established as part of the characterization of the functional Minkowski vectors in our prior joint work with F. Mussnig. The kinematic formula is obtained by integrating these valuations against the motion-invariant measure, so the properties transfer directly. To address the request for an explicit check, we will add a short remark in the introduction of the revised manuscript illustrating the formula for quadratic convex functions, where the relevant measures reduce to explicit expressions involving the Hessian and the integration over the motion group can be carried out by direct computation. revision: yes

  2. Referee: [Proof of the kinematic formula] For lower-semicontinuous convex functions that are not strictly convex, the Monge-Ampère measures may fail to be additive under Minkowski addition in the precise sense required to interchange the integral over the motion group with the vector-valued valuation. The manuscript should either add an explicit continuity or approximation argument or restrict the domain to a subclass where the interchange is justified.

    Authors: The characterization in the joint work with Mussnig is stated for the full space of lower-semicontinuous convex functions and already incorporates the continuity and approximation properties needed to justify the interchange of the valuation with the integral. Nevertheless, we agree that making the approximation step explicit in the present setting would strengthen the exposition. In the revised manuscript we will insert a short paragraph in the proof section outlining an approximation of general convex functions by smooth strictly convex functions (via infimal convolution) and invoking the continuity of the mixed Monge-Ampère measures to pass to the limit. revision: yes

Circularity Check

1 steps flagged

Kinematic formula depends on prior self-cited characterization of functional Minkowski vectors and mixed Monge-Ampère additivity

specific steps
  1. self citation load bearing [Abstract]
    "These vectors, recently introduced and characterized by the author and F. Mussnig, form a natural family of vector-valued valuations on the space of convex functions. This result represents the first integral geometric application of this characterization."

    The manuscript defines the functional Minkowski vectors via the prior joint characterization and then claims the additive kinematic formula follows directly from the mixed Monge-Ampère measures' additivity and continuity. Because those measure properties are taken from the overlapping-author prior work without re-derivation or external verification here, the kinematic identity is load-bearing on the self-citation.

full rationale

The paper's central result is framed as the first integral-geometric application of a characterization established in joint prior work with F. Mussnig. The derivation invokes that characterization to define the vectors and to assert that the mixed Monge-Ampère measures inherit the required additivity and continuity for the kinematic identity to hold. No independent verification or low-dimensional check of these properties in the kinematic setting appears in the provided text, so the new formula reduces to an application of the self-cited premises.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the prior characterization of functional Minkowski vectors (cited but not re-proved) and on standard properties of mixed Monge-Ampère measures; no free parameters or new invented entities are indicated in the abstract.

axioms (2)
  • domain assumption Functional Minkowski vectors satisfy the vector-valued valuation properties established in the recent characterization by the author and F. Mussnig.
    Invoked as the foundation for applying the kinematic formula.
  • standard math Mixed Monge-Ampère measures are additive and continuous in the appropriate topology on convex functions.
    Required to derive the additive kinematic identity.

pith-pipeline@v0.9.0 · 5558 in / 1289 out tokens · 47602 ms · 2026-05-21T23:08:32.022214+00:00 · methodology

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Reference graph

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29 extracted references · 29 canonical work pages · 1 internal anchor

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    If s ≤ t, then (µ w s + λ v t)(x) =    0 if |x| ≤ s and xn ≥ 0 µ(|x| − s) if s < |x| ≤ t and xn ≥ 0 µ(|x| − s) + λ(|x| − t) if |x| > t and xn ≥ 0 0 if |x|n−1 ≤ s, |x| ≤ t and xn < 0 λ(|x| − t) if |x|n−1 ≤ s, |x| > t and xn < 0 µ(|x|n−1 − s) if |x|n−1 > s, |x| ≤ t and xn < 0 µ(|x|n−1 − s) + λ(|x| − t) if |x|n−1 > s, |x| > t and xn < 0

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    To prove Proposition A.3, an essential tool will be the disintegration theorem

    If s > t , then (µ w s + λ v t)(x) =    0 if |x| ≤ t and xn ≥ 0 λ(|x| − t) if t < |x| ≤ s and xn ≥ 0 µ(|x| − s) + λ(|x| − t) if |x| > s and xn ≥ 0 0 if |x| ≤ t and xn < 0 λ(|x| − t) if |x|n−1 ≤ s, |x| > t and xn < 0 µ(|x|n−1 − s) + λ(|x| − t) if |x|n−1 > s and xn < 0. To prove Proposition A.3, an essential tool will be the disintegrat...

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    n j 1 n j−1X k=1 j k (j − k)sn−k+1λj−kµk Z +∞ t 1 rj−k+1√ r2 − s2 dr + n − 1 j jX k=1 j k sn−k−1λj−kµk Z +∞ t 1 rj−k−1√ r2 − s2 dr # = ωn−1 n − 1

    Let s ≤ t. We will split the domain Rn into several parts. (a) The function µws + λvt is of class C2 on the set A1 = {x ∈ Rn : s < |x| ≤ t, xn ≥ 0}, and the Hessian matrix of µws + λvt at a point x in this set has (n − 1)-eigenvalues equal to µ/|x| and the last eigenvalue is equal to zero. Hence, [D2(µws + λvt)(x)]j = (n−1 j µj |x|j if 0 ≤ j ≤ n − 1 0 if ...

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    Z C1 3 ξ(|x|) xn |x|j dx + Z C2 3 ξ(|x|) xn |x|j dx # = λj n − 1 j

    For t < s , we divide the domain Rn into several parts. (a) The function µws + λvt is of class C2 on the set C1 = {x ∈ Rn : t < |x| ≤ s, xn ≥ 0}, and its Hessian matrix at a point x in this set has (n − 1)-eigenvalues equal to λ/|x| and the last eigenvalue is equal to zero. Hence, [D2(µws + λvt)(x)]j = (n−1 j λj |x|j if 0 ≤ j ≤ n − 1 0 if j = n and for 1 ...

    work page