Generalized Sup-Convolutions

• Generalized sup-convolutions are supremal operators that replace classical quadratic kernels with structure-adapted kernels such as action functionals, convex kernels, or averaging constraints.
• They provide a unified framework for regularization, convexification, and geometric inequality liftings across dynamics, convex analysis, and probability settings.
• Applications range from intrinsic Riemannian smoothing and generalized characteristics in Hamilton–Jacobi theory to probabilistic max-convolutions, yielding precise quantitative bounds.

Generalized sup-convolutions are supremal operators in which the classical quadratic kernel

$$u^\varepsilon(x)=\sup_{y}\Big\{u(y)-\frac{1}{2\varepsilon}\|x-y\|^2\Big\}$$

or ordinary affine averaging is replaced by a problem-adapted action, convex kernel, relation, or aggregation rule. In the sources considered here, this includes the forward Lax–Oleinik operator

$$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$

averaging-based sup-convolutions on compact convex domains, supremum representations

$G(x)=\sup_{y\in Y}\{f(y)+\psi(x,y)\},$

relation-driven functional liftings of geometric inequalities, Riemannian sup–inf regularizations with squared distances, and maximum-based generalized convolutions on probability measures (Cannarsa et al., 2016, Hintum et al., 2020, Correa et al., 2017, Malliaris et al., 21 Aug 2025, Azagra et al., 2014, Jasiulis-Gołdyn et al., 2019).

1. Principal forms and common structure

The literature uses the term in several related senses. In each case, the defining operation is a supremum over admissible configurations, but admissibility is encoded differently: by an action functional, an averaging constraint, a convex kernel, a geometric relation, or an extremal binary operation.

Setting	Defining operator	Structural role
Tonelli Hamilton–Jacobi	$(T_t^+u)(x)=\sup_y\{u(y)-A_t(x,y)\}$	Dynamics-adapted regularization and singularity propagation
Convex-domain averaging	$$(f_1*\cdots*f_n)(z)=\sup\{\frac1n\sum_i f_i(x_i): \frac1n\sum_i x_i=z\}$$	Quantitative convexification
Convex analysis	$G(x)=\sup_{y\in Y}\{f(y)+\psi(x,y)\}$	Subdifferential calculus for supremum functions
Functional geometry	$$\square f(z)=\sup_{x\in C(z)}\mathcal{M}_\alpha^{(t)}(f_1(x_1),\ldots,f_m(x_m))$$	Lifting geometric inequalities to functional inequalities
Riemannian regularization	$f^\mu(x)=\sup_z\{f(z)-\frac1{2\mu}d(x,z)^2\}$	Intrinsic $C^{1,1}$ smoothing
Probability on $P_+$	$$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$0	Extremal aggregation and generalized convolutions

A shared theme is replacement of the Euclidean quadratic kernel by a kernel matching the ambient structure. In Tonelli dynamics, the kernel is the exact action $$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$1. In convex geometry, the constraint is barycentric averaging. In functional liftings, the kernel is replaced by a relation $$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$2 and an aggregator $$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$3. In probability, the supremal mechanism becomes the maximum operation itself. This suggests that “generalized sup-convolution” is best understood as a family of supremal envelopes adapted to a given geometry rather than as a single operator.

2. Tonelli action kernels and generalized characteristics

For autonomous Tonelli systems on $$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$4, the fundamental solution is the minimal action

$$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$5

where $$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$6 is $$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$7, strictly convex in $$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$8, superlinear, and uniformly regular. The associated Hamiltonian is given by the Legendre transform

$$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$9

In this setting, the forward Lax–Oleinik operator

$G(x)=\sup_{y\in Y}\{f(y)+\psi(x,y)\},$0

is the generalized sup-convolution. Its kernel is the exact action of the autonomous Tonelli system rather than $G(x)=\sup_{y\in Y}\{f(y)+\psi(x,y)\},$1. For mechanical systems $G(x)=\sup_{y\in Y}\{f(y)+\psi(x,y)\},$2, the small-time asymptotics

$G(x)=\sup_{y\in Y}\{f(y)+\psi(x,y)\},$3

show that $G(x)=\sup_{y\in Y}\{f(y)+\psi(x,y)\},$4 behaves like classical sup-convolution with a dynamics-adapted correction $G(x)=\sup_{y\in Y}\{f(y)+\psi(x,y)\},$5 (Cannarsa et al., 2016).

The key small-time estimates are semiconvexity in $G(x)=\sup_{y\in Y}\{f(y)+\psi(x,y)\},$6, uniform convexity in $G(x)=\sup_{y\in Y}\{f(y)+\psi(x,y)\},$7, and $G(x)=\sup_{y\in Y}\{f(y)+\psi(x,y)\},$8 regularity of $G(x)=\sup_{y\in Y}\{f(y)+\psi(x,y)\},$9 on cones $(T_t^+u)(x)=\sup_y\{u(y)-A_t(x,y)\}$0. These imply localization of maximizers, uniqueness of the maximizer $(T_t^+u)(x)=\sup_y\{u(y)-A_t(x,y)\}$1 for small $(T_t^+u)(x)=\sup_y\{u(y)-A_t(x,y)\}$2, and semiconcavity of $(T_t^+u)(x)=\sup_y\{u(y)-A_t(x,y)\}$3 with constants scaling like $(T_t^+u)(x)=\sup_y\{u(y)-A_t(x,y)\}$4. For Lipschitz $(T_t^+u)(x)=\sup_y\{u(y)-A_t(x,y)\}$5, the supremum in $(T_t^+u)(x)=\sup_y\{u(y)-A_t(x,y)\}$6 is attained, and every maximizer satisfies $(T_t^+u)(x)=\sup_y\{u(y)-A_t(x,y)\}$7.

This operator yields an intrinsic construction of generalized characteristics for semiconcave viscosity solutions of

$(T_t^+u)(x)=\sup_y\{u(y)-A_t(x,y)\}$8

If $(T_t^+u)(x)=\sup_y\{u(y)-A_t(x,y)\}$9 and $$(f_1*\cdots*f_n)(z)=\sup\{\frac1n\sum_i f_i(x_i): \frac1n\sum_i x_i=z\}$$0 is the unique maximizer of $$(f_1*\cdots*f_n)(z)=\sup\{\frac1n\sum_i f_i(x_i): \frac1n\sum_i x_i=z\}$$1, then $$(f_1*\cdots*f_n)(z)=\sup\{\frac1n\sum_i f_i(x_i): \frac1n\sum_i x_i=z\}$$2 is Lipschitz, $$(f_1*\cdots*f_n)(z)=\sup\{\frac1n\sum_i f_i(x_i): \frac1n\sum_i x_i=z\}$$3, and

$$(f_1*\cdots*f_n)(z)=\sup\{\frac1n\sum_i f_i(x_i): \frac1n\sum_i x_i=z\}$$4

The selected initial direction is canonical: $$(f_1*\cdots*f_n)(z)=\sup\{\frac1n\sum_i f_i(x_i): \frac1n\sum_i x_i=z\}$$5 The global theorem states that if $$(f_1*\cdots*f_n)(z)=\sup\{\frac1n\sum_i f_i(x_i): \frac1n\sum_i x_i=z\}$$6 is globally Lipschitz and semiconcave and $$(f_1*\cdots*f_n)(z)=\sup\{\frac1n\sum_i f_i(x_i): \frac1n\sum_i x_i=z\}$$7, then there exists a generalized characteristic $$(f_1*\cdots*f_n)(z)=\sup\{\frac1n\sum_i f_i(x_i): \frac1n\sum_i x_i=z\}$$8 with $$(f_1*\cdots*f_n)(z)=\sup\{\frac1n\sum_i f_i(x_i): \frac1n\sum_i x_i=z\}$$9 and

$G(x)=\sup_{y\in Y}\{f(y)+\psi(x,y)\}$0

The novelty is not only existence but global propagation of singularities along dynamically selected trajectories.

3. Quantitative convexification on convex domains

On a compact convex domain $G(x)=\sup_{y\in Y}\{f(y)+\psi(x,y)\}$1, bounded measurable functions $G(x)=\sup_{y\in Y}\{f(y)+\psi(x,y)\}$2 admit the unweighted sup-convolution

$G(x)=\sup_{y\in Y}\{f(y)+\psi(x,y)\}$3

For $G(x)=\sup_{y\in Y}\{f(y)+\psi(x,y)\}$4, this is written $G(x)=\sup_{y\in Y}\{f(y)+\psi(x,y)\}$5. The operator is equivalent to an infimal convolution by

$G(x)=\sup_{y\in Y}\{f(y)+\psi(x,y)\}$6

Its natural convexification target is the upper convex hull

$G(x)=\sup_{y\in Y}\{f(y)+\psi(x,y)\}$7

The hypograph identities

$G(x)=\sup_{y\in Y}\{f(y)+\psi(x,y)\}$8

connect sup-convolution to Minkowski averaging of sets (Hintum et al., 2020).

For $G(x)=\sup_{y\in Y}\{f(y)+\psi(x,y)\}$9, the central sharp $$\square f(z)=\sup_{x\in C(z)}\mathcal{M}_\alpha^{(t)}(f_1(x_1),\ldots,f_m(x_m))$$0 inequality is

$$\square f(z)=\sup_{x\in C(z)}\mathcal{M}_\alpha^{(t)}(f_1(x_1),\ldots,f_m(x_m))$$1

with exact constants

$$\square f(z)=\sup_{x\in C(z)}\mathcal{M}_\alpha^{(t)}(f_1(x_1),\ldots,f_m(x_m))$$2

These are sharp. For fixed $$\square f(z)=\sup_{x\in C(z)}\mathcal{M}_\alpha^{(t)}(f_1(x_1),\ldots,f_m(x_m))$$3, the paper also proves $$\square f(z)=\sup_{x\in C(z)}\mathcal{M}_\alpha^{(t)}(f_1(x_1),\ldots,f_m(x_m))$$4. In the two-function case,

$$\square f(z)=\sup_{x\in C(z)}\mathcal{M}_\alpha^{(t)}(f_1(x_1),\ldots,f_m(x_m))$$5

and the constant $$\square f(z)=\sup_{x\in C(z)}\mathcal{M}_\alpha^{(t)}(f_1(x_1),\ldots,f_m(x_m))$$6 is sharp.

The proof reduces general convex domains to simplices and then decomposes the simplex into scaled hypersimplices

$$\square f(z)=\sup_{x\in C(z)}\mathcal{M}_\alpha^{(t)}(f_1(x_1),\ldots,f_m(x_m))$$7

The subdivision

$$\square f(z)=\sup_{x\in C(z)}\mathcal{M}_\alpha^{(t)}(f_1(x_1),\ldots,f_m(x_m))$$8

encodes how many vertices of the simplex are used in an $$\square f(z)=\sup_{x\in C(z)}\mathcal{M}_\alpha^{(t)}(f_1(x_1),\ldots,f_m(x_m))$$9-average. The additional concept of $f^\mu(x)=\sup_z\{f(z)-\frac1{2\mu}d(x,z)^2\}$0-averageable sets turns these cells into exact integral estimates. Extremizers are obtained by taking $f^\mu(x)=\sup_z\{f(z)-\frac1{2\mu}d(x,z)^2\}$1 a simplex and $f^\mu(x)=\sup_z\{f(z)-\frac1{2\mu}d(x,z)^2\}$2 equal to the indicator on the vertices of $f^\mu(x)=\sup_z\{f(z)-\frac1{2\mu}d(x,z)^2\}$3. For $f^\mu(x)=\sup_z\{f(z)-\frac1{2\mu}d(x,z)^2\}$4, the required $f^\mu(x)=\sup_z\{f(z)-\frac1{2\mu}d(x,z)^2\}$5-averageability of $f^\mu(x)=\sup_z\{f(z)-\frac1{2\mu}d(x,z)^2\}$6 remains open, which is the stated obstruction to sharp constants in higher dimension.

4. Subdifferential calculus for supremal kernels

In convex analysis, generalized sup-convolutions arise from supremum functions

$f^\mu(x)=\sup_z\{f(z)-\frac1{2\mu}d(x,z)^2\}$7

with $f^\mu(x)=\sup_z\{f(z)-\frac1{2\mu}d(x,z)^2\}$8 for each $f^\mu(x)=\sup_z\{f(z)-\frac1{2\mu}d(x,z)^2\}$9. The issue is to characterize $C^{1,1}$0 from the subdifferentials of the kernels $C^{1,1}$1. The relevant active sets are

$C^{1,1}$2

The starting point is an $C^{1,1}$3-subdifferential formula for $C^{1,1}$4: $C^{1,1}$5 The Banach-space and locally convex extensions replace $C^{1,1}$6 by exact subgradients at nearby points, controlled by norm or seminorm proximity, small function-value gaps, and small dual pairings. In a locally convex space, one such enlargement is

$C^{1,1}$7

leading to formulas for $C^{1,1}$8 that do not require continuity at the reference point (Correa et al., 2017).

When the supremum is finite and continuous at some point of its domain, the formulas simplify. In particular, the paper derives

$C^{1,1}$9

which yields Valadier’s formula when $P_+$0 is continuous at $P_+$1: $P_+$2

Applied to $P_+$3, these formulas show that the subdifferential is assembled from nearby active kernels. If the supremum is attained at a unique $P_+$4, then

$P_+$5

up to the domain normal cone when constraints are present. The classical semiconvex sup-convolution on a Hilbert space,

$P_+$6

fits this framework after the shift

$P_+$7

so that

$P_+$8

whenever the active set description applies.

5. Functional liftings of geometric inequalities

A broader formalism replaces kernels by measurable relations. Given measurable spaces $P_+$9, $$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$00, a relation $$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$01, nonnegative measurable $$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$02, and an aggregator $$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$03, the generalized supremal aggregation is

$$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$04

In the main Euclidean formulation, with fibers $$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$05 and weighted $$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$06-means $$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$07, the paper writes

$$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$08

The operator is required to satisfy monotonicity, superadditivity on sliced functions, and measurability on step inputs (Malliaris et al., 21 Aug 2025).

The core lifting theorem states that, under the paper’s analytic-set and measure-space hypotheses, a geometric inequality for scaled indicators is equivalent to the corresponding functional inequality for nonnegative $$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$09-functions. Thus indicator-level set operations become the model case from which the full functional inequality is deduced. This framework covers affine constraints, $$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$10-Minkowski constraints, linear map constraints from reverse Brascamp–Lieb theory, difference-body constructions, and group-product constraints.

Several applications are developed. From the Gaussian Brunn–Minkowski inequality for origin-symmetric convex sets,

$$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$11

the paper derives a Borell–Brascamp–Lieb statement: if $$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$12 are even unimodal, $$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$13, and

$$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$14

then

$$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$15

The paper also proves that this functional inequality is equivalent to the underlying Gaussian Brunn–Minkowski inequality.

For the log-Brunn–Minkowski conjecture, the generalized sup-convolution uses the relation

$$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$16

and

$$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$17

Indicators recover the set operation exactly: $$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$18 Equivalent functional formulations are also obtained for Barthe’s reverse Brascamp–Lieb inequality and for Schneider’s conjecture on higher-order difference bodies. The framework extends to simply connected nilpotent Lie groups, where the relation is $$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$19. A notable limitation stated in the paper is that the Borell–Ehrhard inequality does not fit the general schema because the required radial convexity fails.

6. Riemannian sup–inf regularization

On a Riemannian manifold $$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$20, the basic envelopes are

$$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$21

and the Lasry–Lions regularization is

$$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$22

This extends classical Euclidean sup–inf convolution to finite- or infinite-dimensional Riemannian manifolds (Azagra et al., 2014).

The main theorem assumes bounded sectional curvature $$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$23, strictly positive injectivity and convexity radii, and bounded uniformly continuous $$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$24. Under the scale restriction $$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$25, the regularizations $$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$26 are uniformly locally semiconvex and uniformly locally semiconcave, hence belong to $$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$27, and converge uniformly to $$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$28 as $$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$29. The paper gives the quantitative bounds

$$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$30

A distinctive feature is the intrinsic definition of global $$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$31, formulated through parallel transport: $$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$32 for nearby $$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$33. The argument depends on curvature-compensated convexity of squared distance combinations and on second-order control of the differential of the exponential map relative to parallel transport.

The regularization preserves order, invariance under isometries, infima, minimizers, and Lipschitzness. When $$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$34, it also preserves local or global convexity. The paper also gives two failure mechanisms: on hyperbolic space, boundedness of $$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$35 is necessary, and if curvature is unbounded then bounded $$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$36 may still produce envelopes that are not $$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$37. This shows that the geometric assumptions are structural rather than technical.

7. Maximum-based generalized convolutions in probability

A different probabilistic strand studies generalized convolutions on $$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$38, the probability measures on $$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$39. The extremal prototype is max-convolution: $$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$40 so that

$$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$41

Here the supremal mechanism is literal: the convolution law is induced by the maximum operation (Jasiulis-Gołdyn et al., 2019).

The central example is the Kendall convolution $$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$42, defined for $$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$43 by

$$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$44

with $$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$45 the Pareto law on $$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$46. For general $$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$47,

$$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$48

Its generalized characteristic function is the Williamson transform

$$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$49

which satisfies the multiplicative identity

$$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$50

The inversion formula is unusually explicit: if $$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$51, then

$$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$52

The paper emphasizes four properties relevant to renewal-type models: monotonicity, existence of a lack-of-memory law, tractable generalized characteristic functions, and convex linear combination structure. For $$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$53, the lack-of-memory distribution is $$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$54 on $$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$55, with density

$$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$56

and the Dirac–Dirac convolution is a two-component mixture. The paper states that Kendall convolution is the only regular generalized convolution with the convex linear combination property for $$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$57; Kendall-type kernels extend this to a three-component setting.

Stochastic representability is another notable feature. The Kucharczak–Urbanik convolution $$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$58 admits an order-statistics representation involving i.i.d. Pareto$$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$59 variables and order statistics of $$(T_t^+u)(x)=\sup_{y\in\mathbb{R}^n}\{u(y)-A_t(x,y)\},$$60 variables. The paper presents this as a route toward extreme value theory and Archimedean copulas. By contrast, Kingman convolution has a Bessel-function transform and rich weak-stability structure, but it is not monotonic and therefore does not play the same role in monotone renewal processes.

Taken together, these strands show that generalized sup-convolutions are not a single theory but a recurrent structural device. Whether expressed through action kernels, barycentric constraints, convex kernels, geometric relations, squared distances, or extremal probability operations, they convert underlying geometry into a supremal envelope whose regularity, duality, or extremal behavior can then be analyzed with quantitative precision.