Generalized Legendre Duality Relation

• Generalized Legendre duality relation is an extension of the classical Legendre–Fenchel correspondence, replacing standard Euclidean pairings with affine deformations, diastasis, and quadratic couplings.
• It unifies diverse frameworks—from convex analysis and Kähler geometry to multi-function inequalities and bundle-theoretic mechanics—while preserving key properties like involution and error decomposition.
• The framework offers practical insights for optimization, thermodynamics, and quantum field theory through both analytic techniques and combinatorial structures.

Searching arXiv for recent and foundational uses of “generalized Legendre duality relation” across geometry, convex analysis, and related areas. Generalized Legendre duality relation denotes a family of extensions of the classical Legendre–Fenchel correspondence, whose standard model is

$$(LF)(\eta):=\sup_{\theta\in\mathbb R^m}\{\langle \theta,\eta\rangle-F(\theta)\},$$

with the biconjugation identity $(F^*)^*=F$ on suitable convex classes. In contemporary work, this pattern is reformulated as affine deformation of convex conjugacy, as diastasis-based duality on Kähler manifolds, as a multi-function quadratic-coupling principle with Gaussian extremizers, as a bundle-theoretic structure on dually flat manifolds and generalized tangent bundles, and as a variational or spectral correspondence in mechanics, thermodynamics, and large-$N$ field theory (Nielsen, 28 Jul 2025, Berndtsson et al., 2016, Nakamura et al., 2024).

1. Convex-analytic archetypes

On $\Gamma_0(\mathbb R^m)$, the Artstein-Avidan–Milman characterization theorem identifies every invertible order-reversing transform with

$$(TF)(\eta)=\lambda (LF)(E\eta+f)+\langle \eta,g\rangle+h, \qquad \lambda>0,\ E\in GL(\mathbb R^m),\ f,g\in\mathbb R^m,\ h\in\mathbb R.$$

Nielsen’s reformulation makes this explicit as ordinary conjugation of an affine-deformed primal function. Writing

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

one has

$L(F_P)=(LF)_{P^\diamond}=F^*_{P^\diamond},$

with

$$P^\diamond= \left( \lambda,\frac{1}{\lambda }A^{-1}, -\frac{1}{\lambda}A^{-1}c, -A^{-1}b, \langle b,A^{-1}c\rangle-d \right), \qquad (P^\diamond)^\diamond=P.$$

In this sense, generalized convex conjugates are convex conjugates of affine-deformed functions rather than a genuinely new conjugacy operation (Nielsen, 28 Jul 2025).

A second convex-analytic line keeps Fenchel–Young duality as the primitive object but reinterprets its gap as an error functional. For admissible primal and dual variables $(\hat U,\hat P)\in\mathcal U\times\mathcal T$, the generalized constitutive relation error is

$$\Psi(\hat U,\hat P)=\phi(\hat U)+\phi^*(\hat P)-\mathcal B(\hat U,\hat P)\ge 0.$$

Its vanishing characterizes exact constitutive compatibility, and for an exact solution pair $(F^*)^*=F$0 the paper proves

$(F^*)^*=F$1

In admissible settings the cross term vanishes, so the duality gap becomes an exact decomposition of primal and dual global errors. The smooth constitutive law $(F^*)^*=F$2 is thereby generalized to the subdifferential relation $(F^*)^*=F$3 (Guo et al., 2016).

2. Kähler and complex-geometric duality

A major geometric extension replaces the Euclidean pairing by Calabi’s diastasis on a compact real analytic Kähler manifold $(F^*)^*=F$4. If $(F^*)^*=F$5 denotes the diastasis, the global generalized Legendre transform is

$(F^*)^*=F$6

Locally this is the coordinate-free version of the polarized transform built from the analytic continuation of a real analytic Kähler potential. The flat model $(F^*)^*=F$7 gives $(F^*)^*=F$8, so the construction reduces to the complexified classical Legendre transform on $(F^*)^*=F$9 (Berndtsson et al., 2016).

For $N$0 sufficiently small in $N$1, the supremum is attained uniquely at a point $N$2, producing a generalized gradient map $N$3. The transform satisfies

$N$4

and the Kähler-form transformation law

$N$5

Consequently $N$6 is a local involutive isometry of the Mabuchi metric

$N$7

so the generalized duality preserves both the infinite-dimensional Riemannian structure on $N$8 and the complex Monge–Ampère measure (Berndtsson et al., 2016).

Lempert’s subsequent analysis places this transform in the symmetric-space picture of the space of Kähler potentials. If $N$9 and $\Gamma_0(\mathbb R^m)$0 is real analytic, then there is a neighborhood $\Gamma_0(\mathbb R^m)$1 and a $\Gamma_0(\mathbb R^m)$2 diffeomorphism $\Gamma_0(\mathbb R^m)$3 such that

$\Gamma_0(\mathbb R^m)$4

and $\Gamma_0(\mathbb R^m)$5 is an isometry of the Mabuchi metric. The converse is equally sharp: if such a $\Gamma_0(\mathbb R^m)$6 symmetry exists about $\Gamma_0(\mathbb R^m)$7, then $\Gamma_0(\mathbb R^m)$8 is real analytic. Real analyticity is therefore both the enabling condition for the diastasis construction and the rigidity criterion for fixed points of complex Legendre duality (Lempert, 2017).

3. Multi-function duality, inverse Brascamp–Lieb theory, and Gaussian saturation

A further extension replaces the two-function pairing by a multi-function quadratic-coupling relation. For an orthogonal decomposition $\Gamma_0(\mathbb R^m)$9, positive weights $$(TF)(\eta)=\lambda (LF)(E\eta+f)+\langle \eta,g\rangle+h, \qquad \lambda>0,\ E\in GL(\mathbb R^m),\ f,g\in\mathbb R^m,\ h\in\mathbb R.$$0, and a symmetric matrix $$(TF)(\eta)=\lambda (LF)(E\eta+f)+\langle \eta,g\rangle+h, \qquad \lambda>0,\ E\in GL(\mathbb R^m),\ f,g\in\mathbb R^m,\ h\in\mathbb R.$$1, the generalized duality relation is

$$(TF)(\eta)=\lambda (LF)(E\eta+f)+\langle \eta,g\rangle+h, \qquad \lambda>0,\ E\in GL(\mathbb R^m),\ f,g\in\mathbb R^m,\ h\in\mathbb R.$$2

This contains the classical two-function inequality $$(TF)(\eta)=\lambda (LF)(E\eta+f)+\langle \eta,g\rangle+h, \qquad \lambda>0,\ E\in GL(\mathbb R^m),\ f,g\in\mathbb R^m,\ h\in\mathbb R.$$3 and the Kolesnikov–Werner many-function relation as special cases (Nakamura et al., 2024).

The principal theorem states that for nonnegative even $$(TF)(\eta)=\lambda (LF)(E\eta+f)+\langle \eta,g\rangle+h, \qquad \lambda>0,\ E\in GL(\mathbb R^m),\ f,g\in\mathbb R^m,\ h\in\mathbb R.$$4 satisfying the generalized relation,

$$(TF)(\eta)=\lambda (LF)(E\eta+f)+\langle \eta,g\rangle+h, \qquad \lambda>0,\ E\in GL(\mathbb R^m),\ f,g\in\mathbb R^m,\ h\in\mathbb R.$$5

where $$(TF)(\eta)=\lambda (LF)(E\eta+f)+\langle \eta,g\rangle+h, \qquad \lambda>0,\ E\in GL(\mathbb R^m),\ f,g\in\mathbb R^m,\ h\in\mathbb R.$$6 and the supremum is over Gaussian tuples satisfying the same quadratic-coupling constraint. The Gaussian admissibility condition is

$$(TF)(\eta)=\lambda (LF)(E\eta+f)+\langle \eta,g\rangle+h, \qquad \lambda>0,\ E\in GL(\mathbb R^m),\ f,g\in\mathbb R^m,\ h\in\mathbb R.$$7

The proof proceeds through a degenerate inverse Brascamp–Lieb inequality, and the central structural result is centered Gaussian saturation: for a large family of degenerate Brascamp–Lieb data, the optimal constant is already attained by centered Gaussian inputs when the functions are even and log-concave (Nakamura et al., 2024).

This framework yields several consequences. It proves the principal Kolesnikov–Werner conjectural case of a Blaschke–Santaló inequality for multiple even functions and multiple symmetric convex bodies, and it also gives a Talagrand-type inequality for multiple even probability measures involving the Wasserstein barycenter. In this setting, generalized Legendre duality is no longer a transform of one function into one dual function; it becomes an inequality class controlled by a quadratic form and solved by a finite-dimensional Gaussian optimization problem (Nakamura et al., 2024).

4. Generalized tangent bundles and algebroid mechanics

In generalized Lie algebroid geometry, Legendre duality is formulated between a vector bundle $$(TF)(\eta)=\lambda (LF)(E\eta+f)+\langle \eta,g\rangle+h, \qquad \lambda>0,\ E\in GL(\mathbb R^m),\ f,g\in\mathbb R^m,\ h\in\mathbb R.$$8 and its dual $$(TF)(\eta)=\lambda (LF)(E\eta+f)+\langle \eta,g\rangle+h, \qquad \lambda>0,\ E\in GL(\mathbb R^m),\ f,g\in\mathbb R^m,\ h\in\mathbb R.$$9, together with their $$F_P(\theta):=\lambda\,F(A\theta+b)+\langle \theta,c\rangle+d,$$0-generalized tangent bundles. A Lagrangian $$F_P(\theta):=\lambda\,F(A\theta+b)+\langle \theta,c\rangle+d,$$1 defines the Legendre bundle morphism

$$F_P(\theta):=\lambda\,F(A\theta+b)+\langle \theta,c\rangle+d,$$2

while a Hamiltonian $$F_P(\theta):=\lambda\,F(A\theta+b)+\langle \theta,c\rangle+d,$$3 defines

$$F_P(\theta):=\lambda\,F(A\theta+b)+\langle \theta,c\rangle+d,$$4

Locally, when regularity holds, the identities

$$F_P(\theta):=\lambda\,F(A\theta+b)+\langle \theta,c\rangle+d,$$5

express generalized Lagrangian–Hamiltonian inversion, and the induced generating functions satisfy

$$F_P(\theta):=\lambda\,F(A\theta+b)+\langle \theta,c\rangle+d,$$6

The essential innovation is the $$F_P(\theta):=\lambda\,F(A\theta+b)+\langle \theta,c\rangle+d,$$7-tangent lift of these bundle morphisms, which allows the transfer of generalized Lie algebroid structures, adapted $$F_P(\theta):=\lambda\,F(A\theta+b)+\langle \theta,c\rangle+d,$$8-bases, distinguished linear $$F_P(\theta):=\lambda\,F(A\theta+b)+\langle \theta,c\rangle+d,$$9-connections, semisprays, and Poincaré–Cartan forms (Arcuş, 2011).

The supplementary treatment of vertical and complete lifts makes this correspondence explicit at the level of lifted sections. On $L(F_P)=(LF)_{P^\diamond}=F^*_{P^\diamond},$0, a section $L(F_P)=(LF)_{P^\diamond}=F^*_{P^\diamond},$1 has generalized vertical and complete lifts

$L(F_P)=(LF)_{P^\diamond}=F^*_{P^\diamond},$2

and

$L(F_P)=(LF)_{P^\diamond}=F^*_{P^\diamond},$3

with dual formulas on $L(F_P)=(LF)_{P^\diamond}=F^*_{P^\diamond},$4. The Legendre-type duality relation then requires the tangent Legendre maps to intertwine these lifts. In its strongest form, both tangent Legendre maps are Lie algebroid morphisms, and $L(F_P)=(LF)_{P^\diamond}=F^*_{P^\diamond},$5 and $L(F_P)=(LF)_{P^\diamond}=F^*_{P^\diamond},$6 are declared Legendre $L(F_P)=(LF)_{P^\diamond}=F^*_{P^\diamond},$7-equivalent (Peyghan et al., 2014).

5. Frobenius manifolds, mirror symmetry, and bundle-theoretic reformulations

Within the theory of WDVV equations and Frobenius manifolds, the classical flat Legendre generator is replaced by a non-flat Legendre field $L(F_P)=(LF)_{P^\diamond}=F^*_{P^\diamond},$8 satisfying

$L(F_P)=(LF)_{P^\diamond}=F^*_{P^\diamond},$9

This condition ensures that the transformed metric and connection

$$P^\diamond= \left( \lambda,\frac{1}{\lambda }A^{-1}, -\frac{1}{\lambda}A^{-1}c, -A^{-1}b, \langle b,A^{-1}c\rangle-d \right), \qquad (P^\diamond)^\diamond=P.$$0

preserve torsion-freeness, metricity, and flatness. On the almost-dual side the induced symmetry is generated not by $$P^\diamond= \left( \lambda,\frac{1}{\lambda }A^{-1}, -\frac{1}{\lambda}A^{-1}c, -A^{-1}b, \langle b,A^{-1}c\rangle-d \right), \qquad (P^\diamond)^\diamond=P.$$1 but by the twisted Legendre field

$$P^\diamond= \left( \lambda,\frac{1}{\lambda }A^{-1}, -\frac{1}{\lambda}A^{-1}c, -A^{-1}b, \langle b,A^{-1}c\rangle-d \right), \qquad (P^\diamond)^\diamond=P.$$2

yielding the commuting diagram between Legendre transformation and almost-duality. This is the mechanism producing explicit maps between rational and trigonometric solutions of WDVV (Strachan et al., 2016).

In mirror symmetry, the smooth Legendre transform of the semi-flat SYZ picture degenerates to the discrete Legendre transform on tropical data. A tropical manifold $$P^\diamond= \left( \lambda,\frac{1}{\lambda }A^{-1}, -\frac{1}{\lambda}A^{-1}c, -A^{-1}b, \langle b,A^{-1}c\rangle-d \right), \qquad (P^\diamond)^\diamond=P.$$3 is sent to

$$P^\diamond= \left( \lambda,\frac{1}{\lambda }A^{-1}, -\frac{1}{\lambda}A^{-1}c, -A^{-1}b, \langle b,A^{-1}c\rangle-d \right), \qquad (P^\diamond)^\diamond=P.$$4

and the local support-function formula

$$P^\diamond= \left( \lambda,\frac{1}{\lambda }A^{-1}, -\frac{1}{\lambda}A^{-1}c, -A^{-1}b, \langle b,A^{-1}c\rangle-d \right), \qquad (P^\diamond)^\diamond=P.$$5

replaces smooth gradient duality. The discrete transform exchanges polytopes and fans, cone and fan pictures of degenerations, and it governs the reconstruction of mirror Landau–Ginzburg potentials. In this setting, generalized Legendre duality is a tropical and combinatorial shadow of the smooth SYZ transform (Ruddat, 2012).

A bundle-theoretic synthesis is proposed in the notion of a Legendre bundle

$$P^\diamond= \left( \lambda,\frac{1}{\lambda }A^{-1}, -\frac{1}{\lambda}A^{-1}c, -A^{-1}b, \langle b,A^{-1}c\rangle-d \right), \qquad (P^\diamond)^\diamond=P.$$6

The symmetric pairing is isotropic on each summand and equals canonical evaluation on mixed terms, while the Legendre morphism

$$P^\diamond= \left( \lambda,\frac{1}{\lambda }A^{-1}, -\frac{1}{\lambda}A^{-1}c, -A^{-1}b, \langle b,A^{-1}c\rangle-d \right), \qquad (P^\diamond)^\diamond=P.$$7

encodes the Hessian metric $$P^\diamond= \left( \lambda,\frac{1}{\lambda }A^{-1}, -\frac{1}{\lambda}A^{-1}c, -A^{-1}b, \langle b,A^{-1}c\rangle-d \right), \qquad (P^\diamond)^\diamond=P.$$8 and the dual flat geometry. This formalism is equivalent to dually flat geometry, carries a canonical para-Kähler structure, and is realized both by exponential families in information geometry and by Hessian QFTs viewed as formal deformations over a parameter $$P^\diamond= \left( \lambda,\frac{1}{\lambda }A^{-1}, -\frac{1}{\lambda}A^{-1}c, -A^{-1}b, \langle b,A^{-1}c\rangle-d \right), \qquad (P^\diamond)^\diamond=P.$$9 (Combe et al., 6 Apr 2026).

6. Thermodynamic, spectral, and statistical-mechanical reinterpretations

In nonequilibrium transport networks, generalized Legendre duality is organized around conjugate branch variables $(\hat U,\hat P)\in\mathcal U\times\mathcal T$0 and $(\hat U,\hat P)\in\mathcal U\times\mathcal T$1 satisfying

$(\hat U,\hat P)\in\mathcal U\times\mathcal T$2

The corresponding transport potentials are the content

$(\hat U,\hat P)\in\mathcal U\times\mathcal T$3

and the co-content

$(\hat U,\hat P)\in\mathcal U\times\mathcal T$4

and at the operating point they satisfy the exact Legendre relation

$(\hat U,\hat P)\in\mathcal U\times\mathcal T$5

The combined Gyarmati Lagrangian

$(\hat U,\hat P)\in\mathcal U\times\mathcal T$6

vanishes on the constitutive manifold and is interpreted as a nonequilibrium analogue of thermodynamic availability. In this framework, extremizations of generalized dissipation and entropy production appear only as special cases, requiring power-law resistances and, for entropy production, isothermal conditions (Porporato et al., 9 Jun 2025).

In thermal large-$(\hat U,\hat P)\in\mathcal U\times\mathcal T$7 quantum field theory, the same phrase is used for a spectral correspondence between retarded correlators in Legendre-transformed or double-trace-deformed theories. For a meromorphic correlator $(\hat U,\hat P)\in\mathcal U\times\mathcal T$8, with poles $(\hat U,\hat P)\in\mathcal U\times\mathcal T$9 and zeros $$\Psi(\hat U,\hat P)=\phi(\hat U)+\phi^*(\hat P)-\mathcal B(\hat U,\hat P)\ge 0.$$0, the spectral product

$$\Psi(\hat U,\hat P)=\phi(\hat U)+\phi^*(\hat P)-\mathcal B(\hat U,\hat P)\ge 0.$$1

satisfies

$$\Psi(\hat U,\hat P)=\phi(\hat U)+\phi^*(\hat P)-\mathcal B(\hat U,\hat P)\ge 0.$$2

At fixed points related by a Legendre transform of generating functionals, the two-point functions obey

$$\Psi(\hat U,\hat P)=\phi(\hat U)+\phi^*(\hat P)-\mathcal B(\hat U,\hat P)\ge 0.$$3

so poles of one correlator become zeros of the other. The generalized relation is therefore a spectral incarnation of functional Legendre duality rather than a new transform on convex functions (Grozdanov et al., 22 Sep 2025).

A statistical-mechanical variant appears in the Legendre duality between spherical and Gaussian spin glasses. Writing $$\Psi(\hat U,\hat P)=\phi(\hat U)+\phi^*(\hat P)-\mathcal B(\hat U,\hat P)\ge 0.$$4, the Gaussian free energy satisfies

$$\Psi(\hat U,\hat P)=\phi(\hat U)+\phi^*(\hat P)-\mathcal B(\hat U,\hat P)\ge 0.$$5

with inverse formula

$$\Psi(\hat U,\hat P)=\phi(\hat U)+\phi^*(\hat P)-\mathcal B(\hat U,\hat P)\ge 0.$$6

This is a generalized convex-conjugacy relation produced by shell decomposition and an equivalence-of-ensembles argument, with explicit correction terms coming from quartic regularization, quadratic confinement, and radial entropy (Genovese et al., 2014).

7. Recurring structures, hypotheses, and scope

Across these constructions, several structural motifs recur. A generalized Legendre duality relation typically replaces the Euclidean bilinear term by another dualizing device: an affine deformation in convex analysis, Calabi’s diastasis in Kähler geometry, a quadratic coupling $$\Psi(\hat U,\hat P)=\phi(\hat U)+\phi^*(\hat P)-\mathcal B(\hat U,\hat P)\ge 0.$$7 in the multi-function setting, a bundle morphism $$\Psi(\hat U,\hat P)=\phi(\hat U)+\phi^*(\hat P)-\mathcal B(\hat U,\hat P)\ge 0.$$8 in dually flat geometry, or a spectral product built from poles and zeros in thermal QFT. What remains recognizable is a passage between paired descriptions together with some combination of order reversal, involutivity, equality at constitutive compatibility, or preservation of geometric data.

The hypotheses, however, are highly setting-dependent. The affine-deformation theorem is finite-dimensional and formulated on $$\Psi(\hat U,\hat P)=\phi(\hat U)+\phi^*(\hat P)-\mathcal B(\hat U,\hat P)\ge 0.$$9 with $(F^*)^*=F$00 and $(F^*)^*=F$01 (Nielsen, 28 Jul 2025). The Kähler transform is local in the space of potentials and requires a compact real analytic Kähler manifold, with involutivity only on a $(F^*)^*=F$02-small neighborhood of the background metric (Berndtsson et al., 2016). The multi-function quadratic-coupling theorem is proved for even inputs, and Gaussian saturation for the inverse Brascamp–Lieb inequality is established when inputs are even and log-concave (Nakamura et al., 2024). The spectral relation requires meromorphic retarded correlators, the thermal product formula, and a large-$(F^*)^*=F$03 regime with simple poles and zeros (Grozdanov et al., 22 Sep 2025). In nonequilibrium transport, dissipation and entropy-production extremizations are not universal consequences of duality but restricted specializations (Porporato et al., 9 Jun 2025).

Taken together, these works present generalized Legendre duality not as a single theorem with a single ambient category, but as a persistent mathematical pattern: a rule for passing between dual variables, dual functions, dual geometries, or dual spectra while retaining a strong remnant of classical Legendre structure. In some contexts that remnant is strict involution; in others it is affine equivalence to ordinary conjugacy, an exact Fenchel–Young-type identity, a local Mabuchi isometry, Gaussian extremality, or a transfer principle between Lagrangian and Hamiltonian descriptions.