Two Generalization Levels of Fenchel Conjugates

Two Generalization Levels of the Fenchel Conjugate

This is a note proposed by Thu-Le, written by ChatGPT.

The Fenchel conjugate is usually introduced as an operation on extended-real-valued functions,

$$f:X\to\overline{\mathbb R}, \qquad f^*(y)= \sup_{x\in X} \{\langle x,y\rangle-f(x)\}.$$

There is, however, a natural hierarchy behind this construction.

At the first level, the Fenchel conjugate acts on functions.
At the second level, functions are lifted to subsets of $X\times\mathbb R$, where conjugation becomes a set polarity.
At the third level, the scalar fiber $\mathbb R$ is replaced by a vector space $Y$, producing a parametric Fenchel polarity indexed by a dual direction $\eta\in Y^*$.

The three levels are connected by two lifting operations and two lowering operations:

$$\boxed{ \text{Function} \;\rightleftarrows\; \text{Scalar-extended set} \;\rightleftarrows\; \text{Vector-extended set}. }$$

The purpose of this note is to make these relationships explicit.

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1. The Commutative Picture

Let $X$ be a real vector space, with dual $X^*$, and let $Y$ be another real vector space.

Fix a nonzero functional

$$\eta\in Y^*.$$

We consider three levels:

$$\begin{array}{ccccc} \mathcal F(X) & \xrightarrow{\quad \operatorname{epi}\quad} & \mathcal P(X\times\mathbb R) & \xrightarrow{\quad L_\eta\quad} & \mathcal P(X\times Y) \\[2mm] \text{Level 1} && \text{Level 2} && \text{Level 3}. \end{array}$$

Here:

• Level 1 consists of extended-real-valued functions;
• Level 2 consists of subsets of the scalar extension $X\times\mathbb R$;
• Level 3 consists of subsets of the vector extension $X\times Y$.

Each level carries its own Fenchel-type duality:

$$\begin{array}{ccccc} f & \xrightarrow{\operatorname{epi}} & A & \xrightarrow{L_\eta} & E \\ \downarrow {*}&& \downarrow{\star}&& \downarrow{\star_\eta} \\ f^* & \xleftarrow{\operatorname{low}} & A^\star & \xleftarrow{Q_\eta} & E^{\star_\eta}. \end{array}$$

Thus there are:

• two lifts: $\operatorname{epi}$ and $L_\eta$;
• three polar operations: $*$, $\star$, and $\star_\eta$;
• two lowering operations: $\operatorname{low}$ and $Q_\eta$.

The classical Fenchel conjugate is therefore the lowest-dimensional shadow of two successive set-level generalizations.

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2. Three Levels and Two Lifting Operations

2.1 Level 1: functions

Let

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

The first lifting operation is the usual epigraph map

$$\operatorname{epi}: \mathcal F(X) \longrightarrow \mathcal P(X\times\mathbb R),$$

defined by

$$
\operatorname{epi}(f)

{(x,r)\in X\times\mathbb R:
f(x)\le r}.
$$

Thus

$$f \quad\longmapsto\quad \operatorname{epi}(f).$$

This moves us from the function level to the scalar-extended set level.

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2.2 Level 2: scalar-extended sets

The ambient space is now

$$X\times\mathbb R.$$

An arbitrary element is written $(x,r)$.

To pass from the scalar fiber $\mathbb R$ to a vector fiber $Y$, fix

$$0\neq\eta\in Y^*.$$

Define the scalarization map

$$q_\eta:Y\to\mathbb R, \qquad q_\eta(u)=\langle\eta,u\rangle.$$

It induces

$$Q_\eta= \operatorname{id}_X\times q_\eta: X\times Y\to X\times\mathbb R,$$

with

$$Q_\eta(x,u)= (x,\langle\eta,u\rangle).$$

The second lifting operation is the inverse-image map

$$L_\eta= Q_\eta^{-1}.$$

Therefore, for $A\subseteq X\times\mathbb R$,

$$\boxed{ L_\eta(A)= \{(x,u)\in X\times Y: (x,\langle\eta,u\rangle)\in A\}. }$$

In particular,

$$L_\eta(\operatorname{epi}f)= \{(x,u): \langle\eta,u\rangle\ge f(x)\}.$$

We denote this vector lift by

$$E_\eta(f):= L_\eta(\operatorname{epi}f).$$

Hence the two successive lifts are

$$\boxed{ f \overset{\operatorname{epi}}{\longmapsto} \operatorname{epi}f \overset{L_\eta}{\longmapsto} E_\eta(f). }$$

When $Y=\mathbb R$ and $\eta=1$,

$$Q_1=\operatorname{id}, \qquad L_1=\operatorname{id},$$

so Level 3 collapses exactly to Level 2.

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3. Three Fenchel Polarities

3.1 Level 1: Fenchel conjugate

For

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

the Fenchel conjugate is

$$\boxed{ f^*(y)= \sup_{x\in X} \{ \langle x,y\rangle-f(x) \}. }$$

Equivalently,

$$f^*(y)\le s$$

if and only if

$$\langle x,y\rangle \le f(x)+s \qquad \text{for all }x\in X.$$

This inequality already suggests a polarity after introducing the epigraph variable.

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3.2 Level 2: Fenchel polar

Let

$$A\subseteq X\times\mathbb R.$$

Define its Fenchel polar by

$$\boxed{ A^\star= \left\{ (y,s)\in X^*\times\mathbb R: \langle x,y\rangle\le r+s \quad \forall(x,r)\in A \right\}. }$$

The underlying Fenchel orthogonality relation is

$$(x,r)\perp_F(y,s) \quad\Longleftrightarrow\quad\langle x,y\rangle\le r+s.$$

The relation between Levels 1 and 2 is exact:

$$\boxed{ (\operatorname{epi}f)^\star= \operatorname{epi}(f^*). }$$

Indeed,

$$\begin{aligned} (y,s)\in(\operatorname{epi}f)^\star &\iff \langle x,y\rangle\le r+s \quad \forall r\ge f(x) \\ &\iff \langle x,y\rangle\le f(x)+s \quad \forall x \\ &\iff s\ge \sup_x \{\langle x,y\rangle-f(x)\} \\ &\iff s\ge f^*(y). \end{aligned}$$

Thus Fenchel conjugation is precisely the function-level representation of Fenchel polarity.

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3.3 Level 3: parametric Fenchel polar

We now replace the scalar fiber $\mathbb R$ by a vector space $Y$.

Fix

$$0\neq\eta\in Y^*.$$

For

$$E\subseteq X\times Y,$$

define the $\eta$-parametric Fenchel polar by

$$\boxed{ E^{\star_\eta}= \left\{ (y,v)\in X^*\times Y: \langle x,y\rangle \le \langle\eta,u+v\rangle \quad \forall(x,u)\in E \right\}. }$$

Equivalently,

$$\langle x,y\rangle \le \langle\eta,u\rangle + \langle\eta,v\rangle.$$

The corresponding parametric Fenchel orthogonality is

$$(x,u)\perp_{F,\eta}(y,v) \quad\Longleftrightarrow\quad \langle x,y\rangle \le \langle\eta,u+v\rangle.$$

When

$$Y=\mathbb R, \qquad \eta=1,$$

this becomes

$$\langle x,y\rangle\le r+s,$$

which is exactly the Level-2 Fenchel polarity.

Therefore,

$$\boxed{ \star_\eta \overset{Y=\mathbb R,\;\eta=1}{\longrightarrow} \star. }$$

The scalar Fenchel polar is the one-dimensional fiber case of the parametric Fenchel polar.

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4. Two Lowering Operations and the Commutative Diagram

The two lifts admit natural lowering operations.

4.1 From Level 2 to Level 1: the lower-envelope map

For

$$A\subseteq X\times\mathbb R,$$

define

$$\operatorname{low}(A)(x)= \inf\{r:(x,r)\in A\}.$$

For epigraphs,

$$\boxed{\operatorname{low}(\operatorname{epi}f)=f.}$$

Since

$$(\operatorname{epi}f)^\star= \operatorname{epi}(f^*),$$

we immediately obtain

$$\boxed{ \operatorname{low} \circ\star \circ\operatorname{epi}= *. }$$

Thus the Level-1 Fenchel conjugate is recovered exactly by lifting to Level 2, applying the Fenchel polar, and lowering again.

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4.2 From Level 3 to Level 2: scalarization

The lowering map from the vector extension to the scalar extension is

$$Q_\eta(x,u)= (x,\langle\eta,u\rangle).$$

For arbitrary sets one should distinguish the direct image

$$Q_\eta(E)= \{ (x,\langle\eta,u\rangle):(x,u)\in E \}$$

from the inverse-image lift $L_\eta=Q_\eta^{-1}$.

For sets obtained through the lift $L_\eta$, the two operations satisfy

$$\boxed{ Q_\eta L_\eta(A)=A, }$$

because $\eta\neq0$ implies that

$$q_\eta:Y\to\mathbb R$$

is surjective.

Thus Level 2 can be recovered exactly from its Level-3 lift.

More importantly, the polarities commute with scalarization:

$$\boxed{ Q_\eta(E^{\star_\eta})= (Q_\eta E)^\star }$$

for $\eta$-saturated sets, and in particular for every set of the form

$$E=L_\eta(A).$$

Consequently,

$$\boxed{ L_\eta(A)^{\star_\eta}= L_\eta(A^\star). }$$

Taking

$$A=\operatorname{epi}f,$$

gives

$$\boxed{ E_\eta(f)^{\star_\eta}= E_\eta(f^*). }$$

We therefore obtain the complete commuting diagram

$$\boxed{\begin{array}{ccccc} f&\xrightarrow{\operatorname{epi}}&\operatorname{epi}f&\xrightarrow{L_\eta} &E_\eta(f)\\[2mm] \downarrow{*}&&\downarrow{\star}&&\downarrow{\star_\eta}\\[2mm] f^*&\xleftarrow{\operatorname{low}}&\operatorname{epi}f^*&\xleftarrow{Q_\eta}&E_\eta(f^*). \end{array}}$$

The three dualities are therefore related by

$$\boxed{ *= \operatorname{low} \circ \star \circ \operatorname{epi}, }$$

and

$$\boxed{ \star= Q_\eta \circ \star_\eta \circ L_\eta }$$

on the lifted class.

Combining both identities,

$$\boxed{ *= \operatorname{low} \circ Q_\eta \circ \star_\eta \circ L_\eta \circ \operatorname{epi}. }$$

Thus the classical Fenchel conjugate can be recovered from the highest-level parametric polarity through two successive lowering operations.

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Preservation of the Two Sum Structures

Passing from the scalar extension $X\times\mathbb R$ to the vector extension $X\times Y$ does not destroy the distinction between the two natural additive structures.

At the set level, one retains:

• fiber sum, where the spatial coordinate is fixed and the fiber coordinates are added;
• spatial sum, where spatial coordinates are combined while the corresponding fiber information is combined accordingly.

The scalarization map preserves addition because

$$\langle\eta,u+v\rangle= \langle\eta,u\rangle+\langle\eta,v\rangle.$$

Hence the lift and lowering operations are compatible with the additive structure carried by the fibers.

In particular, the characteristic Fenchel exchange between the two sum structures survives the passage

$$X\times\mathbb R \quad\longleftrightarrow\quad X\times Y.$$

Schematically,

$$\boxed{ \text{fiber sum} \quad \overset{\text{Fenchel polarity}}{\longleftrightarrow} \quad \text{spatial sum}, }$$

at both the scalar-extended and vector-extended levels, with the appropriate closure or regularization whenever required.

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Summary

The Fenchel conjugate belongs to a three-level hierarchy:

$$\boxed{ \begin{array}{ccccc} \text{Functions} & \rightleftarrows & \text{Scalar-extended sets} & \rightleftarrows & \text{Vector-extended sets} \\ f && A\subseteq X\times\mathbb R && E\subseteq X\times Y \\ * && \star && \star_\eta . \end{array} }$$

The two lifting operations are

$$\boxed{ \operatorname{epi}, \qquad L_\eta=Q_\eta^{-1}, }$$

while the two lowering operations are

$$\boxed{ \operatorname{low}, \qquad Q_\eta. }$$

They recover the lower-level dualities exactly:

$$\boxed{ *= \operatorname{low} \circ\star\circ\operatorname{epi}, }$$

$$\boxed{ \star= Q_\eta\circ\star_\eta\circ L_\eta, }$$

and therefore

$$\boxed{ *= \operatorname{low} \circ Q_\eta \circ\star_\eta \circ L_\eta \circ\operatorname{epi}. }$$

The classical Fenchel conjugate is therefore not an isolated transformation. It is the function-level shadow of a Fenchel polarity on scalar-extended sets, which is itself the scalar shadow of a parametric Fenchel polarity on vector-extended sets.