Sup/Inf-Convolutions in Functional Analysis

• Sup/inf-convolutions are extremal aggregation operations that combine functions by optimizing over admissible decompositions, widely used in convex and nonsmooth analysis.
• They extend classical convolution concepts through frameworks like Moreau–Yosida regularization, Hopf–Lax semigroups, and Minkowski averaging, linking algebraic and geometric structures.
• Applications include regularization, derivation of subdifferential formulas, and lifting geometric inequalities to functional settings, though limitations arise in higher dimensions and nonconvex cases.

Sup/inf-convolutions are extremal aggregation operations that combine functions by optimizing over admissible decompositions of an argument. In the additive setting on $\mathbb{R}^n$, the inf-convolution is

$$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$

while the corresponding sup-convolution is obtained by replacing $\inf$ with $\sup$; on groups one replaces $x-y$ by $y^{-1}x$, and in barycentric formulations one optimizes over tuples with prescribed average. Across convex analysis, nonsmooth analysis, metric geometry, Hamilton–Jacobi theory, and geometric inequalities, these operations appear as Moreau–Yosida-type regularizations, Hopf–Lax/Lax–Oleinik semigroups, max-plus analogues, and functional extensions of set addition and internal group laws (Bachir, 2015, Nam et al., 2014, Hintum et al., 2020, Malliaris et al., 21 Aug 2025).

1. Definitions and principal variants

The classical additive infimal convolution on a real Banach space $X$ is written in the form

$$(f \,\square\, \varphi)(x) := \inf_{w \in X}\{\,f(w) + \varphi(w - x)\,\},$$

with $f:X\to \overline{\mathbb{R}}$ and $\varphi:X\to[0,\infty)$ (Nam et al., 2014). On $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$0, the standard two-function form is

$$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$1

and a corresponding supremal convolution is

$$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$2

(Duggal et al., 11 Aug 2025). In the paper on integral inequalities and Hamilton–Jacobi equations, the notation

$$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$3

is used, together with the level-sum operations

$$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$4

(Rabier, 2015).

On a metric invariant group $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$5, inf-convolution is defined by

$$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$6

equivalently,

$$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$7

The associated sup-convolution is

$$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$8

with max-plus duality

$$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$9

(Bachir, 2015).

A different but closely related formulation appears on a compact convex domain $\inf$0. For bounded measurable $\inf$1, the sup-convolution is

$\inf$2

and the paper also records the inf-convolution analogue

$\inf$3

(Hintum et al., 2020).

The 2025 lifting framework introduces a more abstract notion of generalized sup-convolution. Given measure spaces $\inf$4 and $\inf$5, a functional $\inf$6 mapping $\inf$7 to $\inf$8 is a generalized sup-convolution if it satisfies monotonicity, superadditivity, and measurability on step functions. A general realization is

$\inf$9

where $\sup$0 and $\sup$1 is a weighted mean (Malliaris et al., 21 Aug 2025).

2. Algebraic and geometric interpretations

A central structural result on metric invariant groups is that the internal law itself can be realized as inf-convolution. If $\sup$2 denotes the Kuratowski embedding, then

$\sup$3

This identifies inf-convolution on function space as an extension of the internal law of $\sup$4: the copy of $\sup$5 inside the function space via $\sup$6 is a subgroup of the monoid of $\sup$7-Lipschitz bounded-below functions, and the monoid law restricts to the original group law (Bachir, 2015).

That same paper develops monoid structures in several natural function spaces. If $\sup$8 is a complete metric invariant group with identity $\sup$9, then $x-y$0 is a commutative monoid with identity $x-y$1 and units $x-y$2. The submonoid

$x-y$3

is dense in suitable metrics $x-y$4 and $x-y$5, and for $x-y$6 the minimizer is unique. The map

$x-y$7

is a surjective, continuous monoid morphism satisfying

$x-y$8

(Bachir, 2015).

The same group-theoretic viewpoint extends to Katetov maps. A function $x-y$9 is Katetov if

$y^{-1}x$0

and $y^{-1}x$1 denotes the set of all Katetov maps with the sup metric

$y^{-1}x$2

If $y^{-1}x$3 is group metric invariant with identity $y^{-1}x$4, then $y^{-1}x$5 is a commutative monoid with identity $y^{-1}x$6, the Kuratowski embedding is compatible with $y^{-1}x$7, and the isometric monoid automorphism group satisfies

$y^{-1}x$8

(Bachir, 2015).

On Banach spaces, the convex Katetov subclass

$y^{-1}x$9

carries an additional cone structure. With scalar multiplication

$X$0

the structure $X$1 is a complete metric convex cone, and the map $X$2 embeds $X$3 isometrically as a Banach space, with

$X$4

(Bachir, 2015).

A geometric interpretation is prominent in the barycentric sup-convolution setting. If $X$5 is bounded on a compact convex $X$6, its upper convex hull $X$7 is the smallest concave function majorizing $X$8, and if

$X$9

then

$$(f \,\square\, \varphi)(x) := \inf_{w \in X}\{\,f(w) + \varphi(w - x)\,\},$$0

Thus sup-convolution corresponds to Minkowski averaging of hypograph slabs (Hintum et al., 2020).

3. Regularization, differentiability, and manifold extensions

Inf-convolution is a standard regularization device. In Banach spaces, the quadratic case

$$(f \,\square\, \varphi)(x) := \inf_{w \in X}\{\,f(w) + \varphi(w - x)\,\},$$1

is a Moreau–Yosida-type smoothing (Nam et al., 2014). More generally, if $$(f \,\square\, \varphi)(x) := \inf_{w \in X}\{\,f(w) + \varphi(w - x)\,\},$$2 is subadditive with $$(f \,\square\, \varphi)(x) := \inf_{w \in X}\{\,f(w) + \varphi(w - x)\,\},$$3, then

$$(f \,\square\, \varphi)(x) := \inf_{w \in X}\{\,f(w) + \varphi(w - x)\,\},$$4

and if $$(f \,\square\, \varphi)(x) := \inf_{w \in X}\{\,f(w) + \varphi(w - x)\,\},$$5 is locally calm at $$(f \,\square\, \varphi)(x) := \inf_{w \in X}\{\,f(w) + \varphi(w - x)\,\},$$6 with constant $$(f \,\square\, \varphi)(x) := \inf_{w \in X}\{\,f(w) + \varphi(w - x)\,\},$$7, then $$(f \,\square\, \varphi)(x) := \inf_{w \in X}\{\,f(w) + \varphi(w - x)\,\},$$8 is locally Lipschitz around any $$(f \,\square\, \varphi)(x) := \inf_{w \in X}\{\,f(w) + \varphi(w - x)\,\},$$9 with constant $f:X\to \overline{\mathbb{R}}$0. Under weak lower semicontinuity and level-boundedness assumptions, $f:X\to \overline{\mathbb{R}}$1 is weakly lower semicontinuous (Nam et al., 2014).

Nguyen Mau Nam and Dang Van Cuong develop generalized differentiation formulas for infimal convolutions in Banach spaces. With projection set

$f:X\to \overline{\mathbb{R}}$2

and

$f:X\to \overline{\mathbb{R}}$3

they show, under coercivity and calmness assumptions at $f:X\to \overline{\mathbb{R}}$4,

$f:X\to \overline{\mathbb{R}}$5

For limiting subdifferentials, if $f:X\to \overline{\mathbb{R}}$6, $f:X\to \overline{\mathbb{R}}$7 is coercive, subadditive, continuous at $f:X\to \overline{\mathbb{R}}$8, and $f:X\to \overline{\mathbb{R}}$9 is lower semicontinuous and Lipschitz on $\varphi:X\to[0,\infty)$0, then

$\varphi:X\to[0,\infty)$1

with equality when $\varphi:X\to[0,\infty)$2 is positively homogeneous and either $\varphi:X\to[0,\infty)$3 is finite-dimensional or $\varphi:X\to[0,\infty)$4 is lower regular at $\varphi:X\to[0,\infty)$5 (Nam et al., 2014).

The earlier nonconvex infimal convolution paper proves parallel formulas in arbitrary normed spaces for Fréchet and Hölder subdifferentials. For

$\varphi:X\to[0,\infty)$6

with coercive kernel $\varphi:X\to[0,\infty)$7 and center-Lipschitz $\varphi:X\to[0,\infty)$8 on $\varphi:X\to[0,\infty)$9, the exact Fréchet formula at contact points $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$00 is

$$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$01

and the Hölder analogue is

$$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$02

These identities unify subdifferential formulas for distance and minimal time functions, including

$$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$03

(Nam, 2013).

Regularization by alternating sup- and inf-convolution extends to Riemannian manifolds. Azagra and Ferrera consider

$$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$04

equivalently,

$$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$05

If the sectional curvature satisfies $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$06, and the injectivity and convexity radii are strictly positive, then every bounded, uniformly continuous $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$07 can be uniformly approximated by globally $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$08 functions $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$09 as $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$10 with $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$11 (Azagra et al., 2014).

This manifold regularization preserves several variational and geometric properties. The correspondence $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$12 preserves invariance by isometries, infima, sets of minimizers, ordering, and local or global Lipschitzness; if one additionally assumes $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$13, it preserves local or global convexity (Azagra et al., 2014).

4. Rearrangement principles and integral inequalities

Several recent results study inf- and sup-convolutions through level-set comparison and integral inequalities. In the rearrangement framework on Polish measure spaces, if $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$14 and $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$15 is the symmetric increasing rearrangement defined by

$$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$16

then under an isoperimetric rearrangement hypothesis the main comparison theorem states

$$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$17

for every real $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$18 (Duggal et al., 11 Aug 2025). In Euclidean space with $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$19, $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$20 convex increasing, and spherical rearrangement, this yields

$$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$21

The same paper derives transform comparisons built from suprema. For

$$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$22

if $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$23 is convex, even, and satisfies $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$24, then

$$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$25

for all $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$26. Important corollaries include the Legendre transform comparison

$$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$27

and the polar transform comparison

$$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$28

(Duggal et al., 11 Aug 2025).

The Orlicz-space paper gives a different family of inequalities for infimal convolution. For Borel measurable, bounded-below $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$29 satisfying $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$30, define

$$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$31

Then for every Young function $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$32,

$$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$33

Although no genuine reverse inequality can hold in full generality, the paper proves the reverse-type estimate

$$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$34

where $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$35 and $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$36 are radial transforms based on enclosing balls of upper level sets (Rabier, 2015).

The sharp $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$37 theory for barycentric sup-convolution is developed in low dimensions by van Hintum, Spink, and Tiba. For bounded measurable $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$38 on compact convex $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$39 with $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$40,

$$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$41

with optimal constants

$$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$42

They also prove an optimal two-function inequality

$$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$43

for $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$44, with equality for the indicator of the vertices of a simplex (Hintum et al., 2020).

These sharp constants arise from a hypersimplex decomposition. The $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$45-th $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$46-dimensional hypersimplex is

$$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$47

and for $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$48 the relevant pieces are proved to be $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$49-averageable: $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$50 in dimension $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$51, $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$52 and $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$53 in dimension $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$54, and $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$55, $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$56, and $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$57 in dimension $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$58 (Hintum et al., 2020).

5. Functional liftings of geometric inequalities

The 2025 generalized sup-convolution framework recasts a broad class of functional inequalities as liftings of geometric set inequalities. The central abstract lifting theorem states that, under the generalized sup-convolution axioms and analytic-set measurability assumptions, a set inequality for scaled indicators,

$$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$59

holds for all scaled indicators if and only if the corresponding functional inequality

$$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$60

holds for all non-negative $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$61 (Malliaris et al., 21 Aug 2025).

A basic example is the multiplicative Prékopa–Leindler-type sup-convolution

$$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$62

which is a generalized sup-convolution. If $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$63 and $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$64, then

$$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$65

so the multiplicative sup-convolution is the exponential dual of classical inf-convolution of potentials (Malliaris et al., 21 Aug 2025).

One application is a Borell–Brascamp–Lieb inequality for Gaussian Brunn–Minkowski. Using the set inequality of Eskenazis and Moschidis for origin-symmetric convex bodies and the lifting theorem, the paper proves: if $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$66 is the standard Gaussian probability measure on $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$67, $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$68, $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$69, and $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$70 are even unimodal, then any Borel measurable $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$71 satisfying

$$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$72

obeys

$$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$73

(Malliaris et al., 21 Aug 2025).

The same framework also gives a functional analog of the log–Brunn–Minkowski conjecture. For even unimodal $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$74, the statement

$$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$75

implies

$$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$76

if and only if the geometric inequality

$$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$77

holds for all even convex bodies $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$78 (Malliaris et al., 21 Aug 2025).

Further applications include the equivalence of Barthe’s reverse Brascamp–Lieb inequality with its geometric set inequality, a functional formulation of Schneider’s conjecture via the sup-convolution

$$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$79

and a nilpotent Lie group Borell–Brascamp–Lieb-type inequality. If $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$80 is a simply connected nilpotent Lie group of dimension $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$81 with Haar measure $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$82, $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$83, and

$$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$84

then

$$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$85

(Malliaris et al., 21 Aug 2025).

6. Applications, examples, and limitations

Several model examples make the abstract constructions explicit. In Euclidean space $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$86 with addition,

$$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$87

and for $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$88, $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$89 one has

$$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$90

More generally, if $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$91 have strong minima at $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$92, then $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$93 (Bachir, 2015). On the Heisenberg group with left-invariant metric,

$$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$94

(Bachir, 2015).

Discrete groups yield min-plus convolutions. For $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$95 with the discrete metric, $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$96 consists of sequences $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$97 with $$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$98, and the monoid law is

$$(f \oplus g)(x)=\inf_{y\in \mathbb{R}^n}[f(y)+g(x-y)],$$99

For $\inf$00 one similarly gets periodic sequences with

$\inf$01

(Bachir, 2015).

In Hamilton–Jacobi theory, inf-convolution appears through Hopf–Lax formulas. For

$\inf$02

with $\inf$03, $\inf$04 convex smooth and superlinear, the solution is

$\inf$05

(Duggal et al., 11 Aug 2025). In the Orlicz-space paper, the Hopf–Lax formula is written as

$\inf$06

and the inequalities for $\inf$07 yield $\inf$08- and Orlicz-space bounds for $\inf$09 (Rabier, 2015). The metric-space regularization

$\inf$10

is explicitly identified as an inf-convolution viewpoint connected to Moreau–Yosida and Hamilton–Jacobi semigroups (Bachir, 2015).

The theory has clear dimensional and structural limitations. Sharp barycentric $\inf$11 constants are proved only for $\inf$12, while for $\inf$13 the conjectured formula remains open and the geometric bottleneck is establishing $\inf$14-averageability for the required hypersimplex pieces (Hintum et al., 2020). Rearrangement-based comparison depends on the availability of an isoperimetric rearrangement; where extremal sets are unknown, only weaker constant-loss comparisons are obtained (Duggal et al., 11 Aug 2025). For the group-monoid picture, cancellation does not generally hold for $\inf$15, and the argmin morphism requires existence and uniqueness of strong minimizers (Bachir, 2015).

Global smoothing by sup-inf convolution on manifolds also requires bounded curvature and bounded data. Azagra and Ferrera give two counterexamples showing that the result completely fails, even for nonflat Cartan–Hadamard manifolds, whenever $\inf$16 or $\inf$17 are not bounded (Azagra et al., 2014). A related limitation in the Orlicz theory is that no genuine reverse inequality for $\inf$18 can hold in full generality; the enclosing-ball transforms $\inf$19 are introduced precisely to recover a dimension-dependent reverse-type estimate (Rabier, 2015).

Taken together, these results place sup/inf-convolutions at the intersection of algebraic extension, geometric averaging, regularization, and functional inequality theory. The common theme is that optimization over decompositions of the argument transfers structure from spaces of points or sets to spaces of functions: from internal group laws to monoids of Katetov maps, from Minkowski addition to sup-convolution inequalities, and from convex/geometric transforms to Hamilton–Jacobi evolution (Bachir, 2015, Malliaris et al., 21 Aug 2025).