Affine-Deformed Convex Conjugates

• Affine-deformed convex conjugates are modified Legendre–Fenchel transforms where affine changes in function arguments yield refined duality relations.
• They extend convex analysis by incorporating affine errors and transformations, supporting rigorous geometric constructions in both discrete and continuous settings.
• These conjugates facilitate novel applications in optimization, information geometry, and affine-invariant physical structures by refining dual relationships.

Affine-deformed convex conjugates are convex conjugation (or duality) operations in convex analysis and geometry modified by affine transformations or errors—either in the function argument, in the domain, in the function value, or in the associated support or normal structures. Such deformations arise throughout convex geometry, optimization, functional analysis, and information geometry, providing a flexible analytical framework and allowing for refined duality relations, geometric constructions, and discrete-continuous analogies.

1. Affine Deformations and Convex Conjugation: General Principles

Convex conjugation typically refers to the Legendre–Fenchel transform for a convex function $F:\mathbb{R}^m\to\mathbb{R}\cup\{\infty\}$,

$$(LF)(\eta) = \sup_{\theta} \big\{ \langle\theta, \eta\rangle - F(\theta) \big\}.$$

Affine deformation modifies either the function or its argument/output by an affine transformation. This can take several equivalent forms:

• Pre-composing the argument: $F_A(\theta) = F(A\theta + b)$ for $A\in GL(\mathbb{R}^m)$, $b\in\mathbb{R}^m$;
• Scaling/Translating the output: $$F_P(\theta) = \lambda F(A\theta + b) + \langle \theta, c\rangle + d$$ with $P = (\lambda, A, b, c, d)$, $\lambda>0$;
• Adding an affine error term: $F(\theta) + \alpha$ or more general perturbations such as $F(\theta) + e(\theta, y)$ for an error function $e$.

A foundational result is that the Fenchel conjugate of an affine-deformed function relates to the conjugate of the original function via explicit transformation of both argument and parameters: $L(F_P)(\eta) = (LF)_{P^\diamond}(\eta)$ where $P^\diamond$ is the dual parameter set, e.g., $$P^\diamond = (\lambda, (1/\lambda)A^{-1}, -(1/\lambda)A^{-1}c, -A^{-1}b, \langle b, A^{-1}c\rangle - d)$$ (Nielsen, 28 Jul 2025). The involutive structure $(P^\diamond)^\diamond = P$ preserves duality.

This shows that any “generalized Legendre transform” which is an order-reversing, invertible transformation on the space of lower semi-continuous, proper convex functions must be an ordinary Legendre transform pre/post-composed with affine deformations (Nielsen, 28 Jul 2025).

2. Discrete and Geometric Manifestations: Affine Properties of Convex Polygons

The theory of affine-deformed convex conjugates has discrete geometric analogues, particularly in the theory of convex polygons and their dual curves (1103.2722). A convex equal-area polygon admits affine-invariant discrete notions:

• Affine normals: $n_i = P_{i-1} + P_{i+1} - 2P_i$;
• Affine curvature: $(n_{i+1} - n_i) = -\mu_{i+1/2} v_{i+1/2}$ where $v_{i+1/2} = P_{i+1} - P_i$;
• Discrete affine evolute: the envelope (or intersection locus) of affine-normal lines through the vertices; this evolute is the combinatorial counterpart of the smooth affine conjugate.

These invariants remain unchanged under global affine transformations, and their evolute gives a natural discrete analogue of an affine-deformed dual. The framework supports results such as:

• The discrete six vertices theorem: every convex equal-area polygon with $n\geq6$ has at least six sextactic edges.
• Discrete affine isoperimetric inequality: $\sum_{i=1}^n \mu_{i+1/2} \leq n^2/(2A)$ (with $A$ the area, $n$ the normalized affine perimeter).

Thus, affine-deformed conjugacy in discrete settings is realized as the passage between the original polygon and its affine evolute, with the invariance of associated quantities under affine transformations.

3. Function-Theoretic and Operator-Theoretic Generalizations

Affine deformation underlies a hierarchy of generalized convex conjugate concepts:

• (e, y)-conjugate (Alizadeh et al., 23 Feb 2024), for a function $f$ and an error $e(x, y)$, is defined by

$$f^{e, y}(x^*) = \sup_x \{\langle x^*, x\rangle - f(x) - e(x, y)\}.$$

For $e\equiv 0$, this is the standard Fenchel conjugate. This (e,y)-formulation is natural for so-called $e$-convex functions (which satisfy a perturbed convexity inequality). The $e$-conjugate retains key duality correspondences, e.g.,

$$x^* \in \partial_e f(x) \iff f^{e,x}(x^*) = \langle x^*, x\rangle - f(x),$$

and supports robust optimality conditions for perturbed optimization.

• Affine isoperimetric constructions (1103.2722): In discrete affine differential geometry, affine length and curvature directly correspond to the structure of convex conjugate relationships between original polygons and their evolutes (duals).
• Monotone (cone-deformed) conjugates (Chen et al., 2020): For functions defined on cones and monotone in the induced partial order, conjugates are taken with the supremum restricted to the cone, and the biconjugacy identity $f = f^{**}$ holds with appropriate structural conditions on the cone (e.g., perfection). This yields representation of $f$ as an upper envelope of affine functions, but with the dual slopes restricted to the dual cone.

4. Unified Algebraic Structure of Affine-Deformed Conjugates

The algebraic formulation encompasses the following essential relationships:

• Affine transformation of the function:

$$F_P(\theta) = \lambda F(A\theta + b) + \langle\theta, c\rangle + d$$

induces a corresponding transformation in its convex conjugate:

$$(LF_P)(\eta) = \lambda (LF)\left( (1/\lambda)A^{-1}(\eta - c) \right) + \langle \eta, -A^{-1}b \rangle + (\langle b, A^{-1}c\rangle - d)$$

(Nielsen, 28 Jul 2025).

• Generalized Legendre transforms: Any invertible, order-reversing transformation agreeing on constants/equality structures with the ordinary Legendre transform must be of this affine-deformed type (Nielsen, 28 Jul 2025).
• Convex set deformations: Affine transformations on the domain (polytopes, cones, etc.) induce corresponding deformations in the support/radial/conjugate function structures (Luan et al., 2017), and the duality in face lattices between a body and its polar is compatible with affine deformation (1107.2319).

5. Geometric and Physical Applications: Affine Spacetimes and Domain Foliations

Affine-deformed convex conjugates play a critical role in modern geometric structures.

• Affine spacetime manifolds (Ablondi, 13 Jun 2025) are quotients of convex domains in $\mathbb{R}^{d+1}$ (invariant under affine-deformed discrete groups) carrying an equiaffine structure and a parallel field of proper convex cones, yielding a causality (timelike structure) in the absence of a metric.
• Support functions and convex conjugates are used to describe the boundary of the convex domain as the graph of a convex conjugate (Legendre transform of the support function), where affine deformation of the group action (by a 1-cocycle) induces affine deformation in the support function, and thus in its conjugate. The Cauchy hypersurfaces foliate the spacetime as level sets of the cosmological time, each given by the graph of a Legendre–Fenchel transform under affine deformation.
• The framework generalizes classical results in Lorentzian and projective geometry to higher dimensions and to more general affine gauges.

6. Information Geometry and Dually Flat Spaces

Affine-deformed convex conjugates are foundational in Hessian information geometry and the paper of dually flat manifolds:

• In these contexts, a convex function $F$ serves as a potential on a manifold $M$ with the corresponding metric and affine connections. The dually flat structure depends on the Legendre transform $F^*$, and affine changes in coordinates are natural due to the invariance of the affine structure under coordinate scaling and translation.
• A generalized Legendre transform, including all affine deformations, corresponds to geometric transformations in choosing coordinate systems and dual potentials on the dually flat space (Nielsen, 28 Jul 2025).

7. Summary Table: Canonical Forms and Duality Relations

Setting	Affine Deformation (Primal)	Convex Conjugate (Dual after Deformation)
Standard Legendre	$F(\theta)$	$$F^*(\eta) = \sup \langle \eta, \theta \rangle - F(\theta)$$
Output scaling/translation	$$F_P(\theta) = \lambda F(A\theta + b) + \langle \theta, c\rangle + d$$	$$L(F_P)(\eta) = \lambda (LF)((1/\lambda)A^{-1}(\eta-c)) + \langle \eta, -A^{-1}b\rangle + (\langle b, A^{-1}c\rangle - d)$$
$e$-convex perturbation	$f(x) + e(x, y)$	$(f+e(\cdot, y))^*(x^*)$
Monotone/cone-restricted	domain restricted to cone $C$	supremum/infs over $C$ or $C^*$
Group action on sets	$A(\text{domain})+b$	support/radial/face structures mapped via dual action

These canonical forms provide a universal mechanism for relating affine deformation in the primal problem to structural changes in the dual. Affine-deformed convex conjugates thus serve as a bridge between structural symmetries, geometric dualities, and generalized variational principles in both finite and infinite dimensions.