Functional Minkowski Vectors

• Functional Minkowski vectors are vector-valued valuations on convex functions that extend classical Minkowski relations using Hessian and Monge–Ampère measures.
• They are constructed with continuity in epi-convergence, dually epi-translation invariance, and rotation equivariance, yielding Hadwiger-type classifications.
• Their integral geometric formulations and kinematic formulas bridge functional valuation theory with classical convex and affine geometry.

In contemporary valuation theory on convex functions, functional Minkowski vectors are vector-valued valuations on spaces of convex functions that extend the classical Minkowski relations from convex bodies to the functional setting. They are defined on $\mathrm{Conv}(\mathbb R^n;\mathbb R)$, or dually on the space $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$ of proper, lower semicontinuous, super-coercive convex functions, and are characterized by continuity with respect to epi-convergence, epi-translation invariance or dual epi-translation invariance, and rotation equivariance. Their modern theory is built from Hessian measures, mixed Monge–Ampère measures, and related transforms, culminating in complete Hadwiger-type classifications and additive kinematic formulas (Mouamine et al., 7 Apr 2025, Mouamine et al., 10 Mar 2025).

1. Foundational definition and ambient spaces

Let $\mathrm{Conv}(\mathbb R^n;\mathbb R)$ denote the space of finite-valued convex functions $v:\mathbb R^n\to\mathbb R$, equipped with epi-convergence; on this space epi-convergence coincides with pointwise convergence. A valuation is a map $Z$ such that, whenever $v\vee w$ and $v\wedge w$ belong again to $\mathrm{Conv}(\mathbb R^n;\mathbb R)$,

$Z(v\vee w)+Z(v\wedge w)=Z(v)+Z(w).$

For vector-valued valuations $$z:\mathrm{Conv}(\mathbb R^n;\mathbb R)\to\mathbb R^n$$, the same identity is understood componentwise. The symmetry conditions relevant to functional Minkowski vectors are dually epi-translation invariance, $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$0 for affine $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$1, and rotation equivariance,

$\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$2

for $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$3, or $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$4 in low dimensions (Mouamine et al., 7 Apr 2025).

Convex conjugation relates this theory to the dual space

$\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$5

Conjugation is a continuous involution between $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$6 and $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$7. On the dual side, the natural symmetry is epi-translation invariance,

$\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$8

for translations $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$9 and constants $\mathrm{Conv}(\mathbb R^n;\mathbb R)$0 (Mouamine et al., 10 Mar 2025).

In this setting, a functional Minkowski vector of degree $\mathrm{Conv}(\mathbb R^n;\mathbb R)$1 is a continuous, dually epi-translation invariant, rotation equivariant valuation

$\mathrm{Conv}(\mathbb R^n;\mathbb R)$2

that is homogeneous of degree $\mathrm{Conv}(\mathbb R^n;\mathbb R)$3. The degree parameter parallels the role of rank and homogeneity in classical Minkowski tensors and intrinsic volumes (Mouamine et al., 7 Apr 2025).

2. Measure-theoretic construction

The basic measures are Hessian and Monge–Ampère measures. For $\mathrm{Conv}(\mathbb R^n;\mathbb R)$4, the subdifferential is

$\mathrm{Conv}(\mathbb R^n;\mathbb R)$5

The Monge–Ampère measure is

$\mathrm{Conv}(\mathbb R^n;\mathbb R)$6

and, if $\mathrm{Conv}(\mathbb R^n;\mathbb R)$7 is $\mathrm{Conv}(\mathbb R^n;\mathbb R)$8,

$\mathrm{Conv}(\mathbb R^n;\mathbb R)$9

The Hessian measures $v:\mathbb R^n\to\mathbb R$0, $v:\mathbb R^n\to\mathbb R$1, satisfy

$v:\mathbb R^n\to\mathbb R$2

in the $v:\mathbb R^n\to\mathbb R$3 regime, where $v:\mathbb R^n\to\mathbb R$4 is the $v:\mathbb R^n\to\mathbb R$5-th elementary symmetric function of the eigenvalues of $v:\mathbb R^n\to\mathbb R$6. The mixed Monge–Ampère measures $v:\mathbb R^n\to\mathbb R$7 are defined by the Steiner formula

$v:\mathbb R^n\to\mathbb R$8

with $v:\mathbb R^n\to\mathbb R$9 (Mouamine et al., 7 Apr 2025).

For $Z$0, the admissible radial densities are

$Z$1

Then the functional Minkowski vector is defined by

$Z$2

For $Z$3, this becomes

$Z$4

An equivalent Monge–Ampère representation is

$Z$5

where

$Z$6

For $Z$7 and $Z$8, $Z$9 is a bijection between $v\vee w$0 and $v\vee w$1 (Mouamine et al., 7 Apr 2025).

A closely related family was introduced in the functional Klain–Schneider framework. For $v\vee w$2 and $v\vee w$3,

$v\vee w$4

extended via Hessian measures; its top-degree member is

$v\vee w$5

where $v\vee w$6 satisfies $v\vee w$7. The same work places these operators beside the analytic moment vector

$v\vee w$8

as the top-degree intrinsic-moment operator (Mouamine et al., 10 Mar 2025).

Family	Formula	Role
$v\vee w$9	$v\wedge w$0	Functional Minkowski vector of degree $v\wedge w$1
$v\wedge w$2	$v\wedge w$3	Top-degree epi-translation invariant operator
$v\wedge w$4	$v\wedge w$5	Analytic counterpart of the moment vector

3. Classification theorems

The decisive structural result is the Vectorial Hadwiger Theorem on convex functions. For $v\wedge w$6, a map

$v\wedge w$7

is a continuous, dually epi-translation invariant, rotation equivariant valuation if and only if there exist unique densities $v\wedge w$8 such that

$v\wedge w$9

For $\mathrm{Conv}(\mathbb R^n;\mathbb R)$0, the same statement holds with $\mathrm{Conv}(\mathbb R^n;\mathbb R)$1-equivariance. Equivalently, there exist unique rotation equivariant vector densities $\mathrm{Conv}(\mathbb R^n;\mathbb R)$2 such that

$\mathrm{Conv}(\mathbb R^n;\mathbb R)$3

and rotation equivariance forces $\mathrm{Conv}(\mathbb R^n;\mathbb R)$4 (Mouamine et al., 7 Apr 2025).

The dual theory on $\mathrm{Conv}(\mathbb R^n;\mathbb R)$5 is formally parallel. Every continuous, epi-translation invariant, rotation equivariant valuation $\mathrm{Conv}(\mathbb R^n;\mathbb R)$6 admits a unique decomposition

$\mathrm{Conv}(\mathbb R^n;\mathbb R)$7

with the same density classes, now represented through the dual mixed Monge–Ampère measures $\mathrm{Conv}(\mathbb R^n;\mathbb R)$8 (Mouamine et al., 7 Apr 2025).

A complementary classification is the vector-valued functional Klain–Schneider theorem. If

$\mathrm{Conv}(\mathbb R^n;\mathbb R)$9

is continuous, dually translation covariant, vertically translation invariant, dually simple, and rotation equivariant, then

$Z(v\vee w)+Z(v\wedge w)=Z(v)+Z(w).$0

for suitable $Z(v\vee w)+Z(v\wedge w)=Z(v)+Z(w).$1 and $Z(v\vee w)+Z(v\wedge w)=Z(v)+Z(w).$2 with $Z(v\vee w)+Z(v\wedge w)=Z(v)+Z(w).$3. Here $Z(v\vee w)+Z(v\wedge w)=Z(v)+Z(w).$4 is translation covariant, while $Z(v\vee w)+Z(v\wedge w)=Z(v)+Z(w).$5 is dually epi-translation invariant. This identifies the top-degree functional Minkowski vector as one of the two fundamental rotation-equivariant building blocks in the simple covariant theory (Mouamine et al., 10 Mar 2025).

These theorems place functional Minkowski vectors within the same formal role that intrinsic volumes and Minkowski tensors occupy in classical valuation theory. A plausible implication is that, once the relevant symmetry class is fixed, the radial density completely determines the operator.

4. Relation to classical Minkowski theory

The most striking contrast with classical convex-body theory is that the new vector-valued valuations have no nontrivial classical convex-body counterparts in the lower degrees. For convex bodies $Z(v\vee w)+Z(v\wedge w)=Z(v)+Z(w).$6, the classical Minkowski relations assert

$Z(v\vee w)+Z(v\wedge w)=Z(v)+Z(w).$7

so all rank-1 Minkowski tensors associated with area or curvature measures vanish; only the moment vector survives in rank $Z(v\vee w)+Z(v\wedge w)=Z(v)+Z(w).$8. Correspondingly, for support functions $Z(v\vee w)+Z(v\wedge w)=Z(v)+Z(w).$9, the family $$z:\mathrm{Conv}(\mathbb R^n;\mathbb R)\to\mathbb R^n$$0 vanishes: $$z:\mathrm{Conv}(\mathbb R^n;\mathbb R)\to\mathbb R^n$$1 This is the precise sense in which functional Minkowski vectors “lack classical counterparts” (Mouamine et al., 10 Mar 2025).

The functional theory bypasses this vanishing through nonconstant densities and through the Monge–Ampère measure of translated support functions. If $$z:\mathrm{Conv}(\mathbb R^n;\mathbb R)\to\mathbb R^n$$2, then $$z:\mathrm{Conv}(\mathbb R^n;\mathbb R)\to\mathbb R^n$$3 is a Dirac measure at $$z:\mathrm{Conv}(\mathbb R^n;\mathbb R)\to\mathbb R^n$$4 scaled by $$z:\mathrm{Conv}(\mathbb R^n;\mathbb R)\to\mathbb R^n$$5, and hence

$$z:\mathrm{Conv}(\mathbb R^n;\mathbb R)\to\mathbb R^n$$6

which is nonzero for $$z:\mathrm{Conv}(\mathbb R^n;\mathbb R)\to\mathbb R^n$$7 and suitable $$z:\mathrm{Conv}(\mathbb R^n;\mathbb R)\to\mathbb R^n$$8. This exhibits genuinely translation-sensitive information that disappears in the classical body setting (Mouamine et al., 10 Mar 2025).

A second bridge to classical geometry is the area-measure representation. For $$z:\mathrm{Conv}(\mathbb R^n;\mathbb R)\to\mathbb R^n$$9 and $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$00, there exists a convex body $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$01 associated with $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$02 such that

$\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$03

where the gnomonic projection from the lower hemisphere is

$\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$04

This formula makes the link to mixed area measures explicit and shows how the functional setting alters the classical Minkowski relations by inserting singular or nonconstant densities (Mouamine et al., 7 Apr 2025).

The elementary examples reflect the same phenomenon. Functional Minkowski vectors vanish on radial functions, including $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$05 and $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$06 with $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$07 positive definite, because the relevant integrands are odd or because of radial symmetry and $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$08-equivariance (Mouamine et al., 7 Apr 2025).

5. Integral geometry, section formulas, and kinematic structure

The first integral-geometric applications appear in two complementary forms. The first is a Kubota-type representation: for $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$09 and $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$10,

$\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$11

where $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$12, $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$13 is the Grassmannian of $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$14-dimensional subspaces, and $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$15 is the Monge–Ampère measure on $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$16. This representation removes singular densities by passing to lower-dimensional sections (Mouamine et al., 7 Apr 2025).

The second is the additive kinematic formula established for the mixed Monge–Ampère form of functional Minkowski vectors. Let $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$17 and let $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$18 satisfy $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$19. Then, for every $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$20,

$\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$21

equals a bilinear sum over $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$22 involving

$\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$23

integrated against $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$24. The scalar companion formula is symmetric in $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$25 and $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$26; the vector-valued formula is rotation invariant in $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$27 but rotation equivariant in $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$28, reflecting the tensorial asymmetry introduced by the factor $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$29 (Mouamine, 11 Aug 2025).

A useful specialization inserts a support function $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$30 and yields

$\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$31

as a sum of intrinsic-volume multiples of the lower-degree operators

$\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$32

This recovers the classical intrinsic volumes $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$33 as structure constants inside the functional theory (Mouamine, 11 Aug 2025).

These formulas supply the integral-geometric analogue of the classification theorems. They show that functional Minkowski vectors are not only classification objects but also natural carriers of additive kinematic information. Open directions explicitly identified in the literature include further characterization of singular densities, sharp inequalities, functional analogs of the Minkowski problem, and extensions to other symmetry groups or function classes (Mouamine et al., 7 Apr 2025).

6. Broader meanings and adjacent frameworks

The phrase “functional Minkowski vectors” also appears in several neighboring literatures, but with technically different meanings. In generalized Minkowski spaces, the underlying object is a gauge $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$34, that is, a positively homogeneous, subadditive functional on a finite-dimensional vector space. There the focus is on orthogonality notions such as Birkhoff and isosceles orthogonality, with characterizations through subdifferentials, directional derivatives, dual gauges, and supporting hyperplanes. In the Finsler-Minkowski and Funk-space literature, vectors are called functional Minkowski vectors when their lengths are governed by a Minkowski functional $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$35 or a base-point-dependent Finsler function $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$36; averaging procedures then produce an averaged Riemannian metric and associated Randers structures (Jahn, 2016, Vincze, 2013).

In anisotropic distance theory for polyhedral gauges, the key “functional Minkowski vector” is $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$37, where $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$38 is the dual gauge and $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$39 is the inward Euclidean normal at a nearest boundary point. The distance-to-boundary function

$\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$40

is the unique viscosity solution of

$\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$41

and satisfies the anisotropic ray formula

$\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$42

whenever $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$43 is differentiable at $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$44 (Safdari, 10 Dec 2025).

A further affine-geometric usage occurs in the discrete functional $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$45 Minkowski problem. For $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$46, the gradients on the affine pieces determine directions $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$47 and weights $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$48, producing a discrete measure

$\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$49

The problem is to find the unique polytope $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$50 whose $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$51 surface area measure equals this functional measure. The resulting operator $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$52 is $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$53-equivariant and an $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$54-Blaschke valuation, and it underlies the general affine Pólya–Szegő principle and affine Sobolev inequalities (Wang, 2020).

The phrase is also used outside convex valuation theory. In invariant special relativity, electric and magnetic fields are treated as observer-dependent Minkowski $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$55-vectors

$\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$56

with Lorentz transformations acting on both $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$57 and the observer velocity $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$58, so that $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$59 and $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$60 transform as genuine $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$61-vectors without mutual mixing (Ivezić, 2022). In polarization optics, the Stokes vector $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$62 is a Minkowski $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$63-vector with invariant

$\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$64

and non-depolarizing Mueller matrices act as Lorentz transformations in Stokes–Minkowski space (Stenflo, 2019). In the Gaussian Brunn–Minkowski setting, a “functional Minkowski vector” is the interpolation path

$\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$65

of coupled random vectors, along which $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$66 is concave for even strongly log-concave laws (Aishwarya et al., 4 Apr 2025). In cosmological data analysis, “Functional Minkowski Vectors” are simply the stacked data vectors of Minkowski functionals evaluated across thresholds and needlet scales for CMB lensing maps (Hamann et al., 2023). In combinatorial polytope theory, the phrase has been used to frame the $\mathrm{Conv}_{\mathrm{sc}}(\mathbb R^n)$67-vector of a Minkowski sum as a function of the summands, leading to exact linear relations among face numbers of partial sums (Weibel, 2010).

This suggests that the expression has become a family-resemblance term rather than a single universal definition. Nevertheless, the valuation-theoretic notion on convex functions remains the most systematic and most fully classified meaning: a canonical family of continuous, rotation-equivariant, epi-translation invariant vector-valued valuations represented through Hessian and Monge–Ampère measures, linked to classical Minkowski relations, and now equipped with section formulas and additive kinematic formulas (Mouamine et al., 7 Apr 2025, Mouamine, 11 Aug 2025).