---
title: Generalized Convex Functions (GCFs)
url: https://www.emergentmind.com/topics/generalized-convex-functions-gcfs
type: topic
---

# Generalized Convex Functions (GCFs)

Generalized convex functions (GCFs) form an umbrella family of non-equivalent extensions of ordinary convexity rather than a single canonical class. In the works surveyed here, the label covers modified Jensen-type inequalities, generalized conjugacies, epigraphic and polyhedral structures, geodesic and connection-dependent convexity, and kernel-generated transforms. What unifies these constructions is that each replaces either the classical interpolation rule \(f(tx+(1-t)y)\le tf(x)+(1-t)f(y)\) or the usual Fenchel pairing by a structure adapted to a different geometry, algebra, or optimization model [1404.3964][2208.06589][2402.15597][1705.06892][2409.14434][2509.04477].

## 1. Scope and conceptual taxonomy

A first point of orientation is terminological. In the surveyed literature, “generalized convexity” may mean an interpolation inequality with altered weights, as in \(s\)-convex, \((\alpha,m)\)-convex, GA-convex, or fractal-set convexity; a path-dependent notion generated by a map \(g\), as in \(X\)-convexity; a conjugacy-based notion, as in Capra-convexity and \((e,y)\)-conjugacy; a structural epigraph class, as in generalized polyhedral convexity; or a geometry-dependent notion, as in g-convexity on manifolds and convexity of order \(\alpha\) in geometric function theory [1308.3954][1511.03308][2010.13323][2112.10181][1603.07116][2303.10520].

Ordinary convexity is recovered in several of these frameworks by specialization. In the fractal-set setting, taking \(\alpha=1\) recovers standard convexity; in \(s\)-\((\alpha,m)\)-convexity, \(\alpha=1\), \(m=1\), \(s=1\) gives ordinary convexity; in \(X\)-convexity, choosing \(g(t)=t\) turns \(\delta(r-t)+g(t)\) into the usual affine combination \(\delta r+(1-\delta)t\); in e-convexity, \(e\equiv 0\) yields the classical convex inequality; and in the kernel-generated theory, \(\Phi(x,y)=\langle x,y\rangle\) recovers Fenchel-Legendre convexity [1404.3964][1308.3954][2208.06589][2402.15597][2509.04477].

## 2. Interpolation-based extensions

One major branch of GCF theory modifies the interpolation rule itself. On the fractal-number set \(\mathbb R^\alpha\), generalized convexity is defined by
\[
f(\lambda x_1+(1-\lambda)x_2)\le \lambda^\alpha f(x_1)+(1-\lambda)^\alpha f(x_2),
\]
with \(0<\alpha\le 1\). In that setting, the paper establishes a three-point slope characterization, equivalence with monotonicity of the local fractional derivative \(f^{(\alpha)}\), a second local fractional derivative criterion \(f^{(2\alpha)}\ge 0\), and generalized Jensen and Hermite–Hadamard inequalities [1404.3964].

A second family uses parametric perturbations of convex weights. The classes \(K^{\alpha,s}_{m,1}\) and \(K^{\alpha,s}_{m,2}\) are defined through \(s\)-\((\alpha,m)\)-convexity in the first and second senses, for example
\[
f(\mu x+(1-\mu)y)\le \mu^{\alpha s}f(x)+m\bigl(1-\mu^{\alpha s}\bigr)f(y/m),
\]
and support weighted Hermite–Hadamard-type estimates with beta-function coefficients [1308.3954]. The later notion of general \(s\)-convexity adds a perturbation map \(\vartheta\) and defines
\[
h(\theta b_1+(1-\theta)b_2)\le \theta^s[h(b_1)+\vartheta(b_1,\theta)]+(1-\theta)^s[h(b_2)+\vartheta(b_2,\theta)]+\vartheta\!\left(\frac{b_1+b_2}{2},\theta\right),
\]
together with closure under positive combinations, maxima, an epigraph characterization via general \(s\)-convex sets, and sufficient optimality conditions including a KKT-type theorem [2301.00649]. In a different direction, \(X\)-convexity replaces the affine segment by \(\delta(r-t)+g(t)\); this extends convexity to certain nonconvex sets, induces quasi-\(X\)-convex, strictly quasi-\(X\)-convex, and semi-strictly quasi-\(X\)-convex variants, and preserves classical local-to-global minimality phenomena under an additional proximity condition [2208.06589].

A third branch replaces arithmetic interpolation by interpolation according to means or by coordinate slicing. GA-convexity requires
\[
f(x^\lambda y^{1-\lambda})\le \lambda f(x)+(1-\lambda)f(y),
\]
and leads to Hermite–Hadamard–Fejér type inequalities via Hadamard fractional integrals, especially when \(|f'|\) or \(|f'|^q\) is GA-convex [1511.03308]. Coordinate-wise convexity, by contrast, is weaker than joint convexity: on a rectangle \(R=[a,b]\times[c,d]\), it means convexity in each variable separately, and supports composite Hermite–Hadamard inequalities indexed by the number of partition subintervals \(n\in\mathbb N\) [1712.10331].

## 3. Conjugacy-based and kernel-based generalized convexity

Another major direction changes the duality pairing rather than the interpolation rule. For functions of the support \(F\circ \operatorname{supp}\), ordinary Fenchel conjugacy is structurally inadequate because such functions are \(0\)-homogeneous. The Capra coupling
\[
\dot c(x,y)=\frac{\langle x,y\rangle}{\|x\|}\quad (x\neq 0),\qquad \dot c(0,y)=0
\]
is constant along primal rays, and Capra-convexity is defined by equality with the Capra biconjugate. Under orthant-strict monotonicity of the source norm and its dual norm, every nondecreasing finite-valued set-function \(F:2^V\to\mathbb R\) yields a Capra-convex support function \(F\circ\operatorname{supp}\). The same paper also proves a hidden-convexity representation \(F\circ\operatorname{supp}=\mathcal L_0^F\circ n\) and an exact variational formulation involving generalized local-\(K\)-support dual norms [2010.13323].

The e-convex framework perturbs convexity by an error function \(e:X\times X\to\mathbb R_+\cup\{+\infty\}\):
\[
f(tx+(1-t)y)\le t f(x)+(1-t)f(y)+t(1-t)e(x,y).
\]
Its central dual object is the \((e,y)\)-conjugate
\[
f^{e,y}(x^*)=\sup_{x\in X}\{\langle x^*,x\rangle-f(x)-e(x,y)\},
\]
which satisfies \(f^{e,y}(x^*)=(f+e(\cdot,y))^*(x^*)\). The e-subdifferential is linked to this conjugacy by the generalized Fenchel equality
\[
x^*\in \partial_e f(x)\iff f^{e,x}(x^*)+f(x)=\langle x^*,x\rangle,
\]
and the framework yields generalized Fermat rules: global minimizers satisfy \(0^*\in \partial_e f(x_0)\), while local minimizers satisfy \(0^*\in \partial_{2e}f(x_0)\). The class is strictly broader than ordinary convexity; for example, \(f(x)=-x^2\) is e-convex for \(e(x,y)=(x-y)^2\) [2402.15597].

A related but more global kernel-based theory takes a surplus \(\Phi:X\times Y\to\mathbb R\) and defines generalized convexity through \(\Phi\)-transforms. A \(Y\)-convex function on \(X\) can be approximated by finitely \(Y\)-convex functions of the form
\[
f(x)=\max_{y\in \tilde Y}\{\Phi(x,y)-r(y)\},
\]
where \(\tilde Y\subseteq Y\) is finite. Under compactness of \(X\) and \(Y\) and local Lipschitz continuity of \(\Phi\), this finite class satisfies a universal approximation property in the uniform norm; under semiconvexity of \(\Phi\), the corresponding gradients also satisfy a universal approximation property. The same representation is used to reduce structured bilevel problems in optimal transport and mechanism design to optimization over a finite-dimensional generalized-convex parameterization [2509.04477].

## 4. Differential and geometric incarnations

In geometric function theory, generalized convexity appears as convexity of order \(\alpha\). The class
\[
\mathcal F(\alpha)=\left\{f\in\mathcal S:\ \Re\!\left(1+\frac{z f''(z)}{f'(z)}\right)>\alpha,\ z\in\mathbb D\right\},\qquad -\frac12\le \alpha<1,
\]
contains the ordinary convex class at \(\alpha=0\), becomes more rigid for \(0<\alpha<1\), and for \(\alpha=-1/2\) consists of functions convex in one direction. A key structural formula is
\[
f'(z)=\int_0^{2\pi}(1-e^{i\theta}z)^{2\alpha-2}\,d\nu(\theta),
\]
from which the coefficient representation
\[
a_n=A_n(\alpha)\int_0^{2\pi}e^{i(n-1)\theta}\,d\nu(\theta)
\]
follows. This makes sharp extremal problems tractable, and the paper solves the generalized Zalcman coefficient problem on \(\mathcal F(\alpha)\) for all \(n\ge 3\), all \(\lambda>0\), and all \(-\frac12\le \alpha<1\), thereby proving the generalized Zalcman conjecture for convex functions of order \(\alpha\) [1603.07116].

A different geometric meaning of generalized convexity arises on manifolds. There, a smooth function is g-convex with respect to a torsion-free affine connection \(\nabla\) if its restriction to every geodesic is convex in the ordinary one-variable sense, equivalently if \(\Hess_\nabla f(x)\succeq 0\) at every point. The paper proves that if a smooth function has no critical points, then one can prescribe its Hessian arbitrarily by a suitable connection, hence make it g-convex. At the same time, strong sparseness results hold: on a compact manifold, the set of g-convex functions with respect to some connection is nowhere dense in \(C^\infty(\mathcal M)\); for generic polynomials on \(\mathbb R^n\), g-convexity under a geodesically complete connection forces at most one critical point; and the density of g-convex univariate, quadratic, monomial, and additively separable polynomials decreases asymptotically to zero in the senses made precise in the paper [2409.14434].

## 5. Generalized polyhedral convexity and multifunctions

Generalized polyhedral convexity is a structural, epigraph-based notion rather than an axiomatic modification of Jensen’s inequality. In a locally convex Hausdorff topological vector space \(X\), a function is generalized polyhedral convex iff its epigraph is a generalized polyhedral convex set. Equivalently, a proper function \(f\) is generalized polyhedral convex iff \(\operatorname{dom}f\) is a generalized polyhedral convex set and
\[
f(x)=
\begin{cases}
\max\{\langle v_k^*,x\rangle+\beta_k\mid k=1,\dots,m\}, & x\in \operatorname{dom}f,\\
+\infty, & x\notin \operatorname{dom}f.
\end{cases}
\]
The same paper proves that proper convex \(f\) is generalized polyhedral convex iff it is generalized piecewise linear, that the conjugate of a proper generalized polyhedral convex function is again proper generalized polyhedral convex, that directional derivatives stay in the same class, and that the infimal convolution of a generalized polyhedral convex function and a polyhedral convex function is a polyhedral convex function [1705.06892].

The multifunction extension replaces epigraphs by graphs. A multifunction \(F:X\rightrightarrows Y\) is generalized polyhedral convex if \(\operatorname{gph}F\) is a generalized polyhedral convex set, equivalently if
\[
\operatorname{gph}F=
\big\{(x,y)\in X\times Y\mid A_1(x)+A_2(y)=z,\ \langle x_i^*,x\rangle+\langle y_i^*,y\rangle\le \beta_i,\ i=1,\dots,m\big\}.
\]
Within this class, each value \(F(x)\) is generalized polyhedral convex, domains and ranges are generalized polyhedral convex under a finite-codimensional closedness assumption on the relevant linear map, and compositions are again generalized polyhedral convex under a corresponding closedness assumption on \((A_2,B_1)\). The paper also studies direct and inverse images and optimal value functions \(\mu(x)=\inf\{\varphi(x,y)\mid y\in F(x)\}\), extending generalized polyhedral convexity from sets and functions to graph-defined set-valued maps [2303.10520].

These structural results feed directly into nonconvex optimization with convex components. In generalized polyhedral DC optimization on locally convex Hausdorff spaces, the objective has the form \(g-h\) with \(g,h\in\Gamma_0(X)\), and generalized polyhedral convexity of one or both components yields much sharper conclusions than arbitrary convex data. If \(h\) is proper generalized polyhedral convex, then at interior points of \(\operatorname{dom}h\) local optimality is equivalent to
\[
\partial h(\bar x)\subset \partial(g+\delta_C)(\bar x).
\]
In the fully generalized polyhedral case, the local solution set is a finite union of semi-closed generalized polyhedral convex sets, the global solution set is a finite union of generalized polyhedral convex sets, and selection-based DCA becomes eventually periodic because the relevant subdifferentials take only finitely many values [2411.19272].

## 6. Optimization-theoretic consequences and recurrent themes

Some generalized-convex classes are designed almost entirely around optimization consequences. In the algebraic framework of a nonempty set \(X\) equipped with a binary operation \(\circ\), a function is \((\circ,p,q)\)-convex if
\[
f(x\circ y)\le p f(x)+q f(y),\qquad p,q>0.
\]
This single inequality subsumes Jensen convexity and subadditivity as special cases. The paper proves a maximum theorem: if \(f_1,\dots,f_n\) are \((\circ,p,q)\)-convex and \(0\le \max(f_1(x),\dots,f_n(x))\) for all \(x\in X\), then there exists \((\lambda_1,\dots,\lambda_n)\in S_n\) such that \(0\le \sum_{i=1}^n\lambda_i f_i(x)\) for all \(x\in X\). From this it derives a generalized Karush–Kuhn–Tucker theorem for the constrained problem \(\min f_0(x)\) subject to \(f_i(x)\le 0\) [2112.10181].

Across the surveyed literature, optimization results repeatedly take the form “generalized convexity restores global structure.” In the \(X\)-convex framework, local minima become global under the proximity condition used in the paper; in general \(s\)-convexity, first-order conditions and KKT-type multipliers become sufficient for global optimality; in e-convexity, conjugacy and e-subdifferentials produce generalized Fermat rules; and in kernel-generated \(Y\)-convexity, optimal transport and mechanism design reduce to optimization over generalized-convex potentials and their gradients [2208.06589][2301.00649][2402.15597][2509.04477].

A recurring source of confusion is that these theories are not interchangeable. Coordinate-wise convexity is weaker than joint convexity, with \(f(x,y)=xy\) on \([0,1]^2\) serving as the standard example; \(X\)-convexity can hold on nonconvex sets such as \(M=[1,2]\cup[3,\infty)\); e-convexity includes genuinely nonconvex functions such as \(-x^2\); generalized polyhedral convexity is a finite-representability property of epigraphs and graphs; and manifold g-convexity depends on the chosen connection or metric rather than on an intrinsic Jensen inequality [1712.10331][2208.06589][2402.15597][1705.06892][2409.14434]. The most accurate general characterization is therefore umbrella-like: generalized convex functions are ordinary convex functions recast through altered interpolation, altered duality, altered geometry, or altered epigraphic structure.

Source: https://www.emergentmind.com/topics/generalized-convex-functions-gcfs