Gconvex: Contextual Generalized Convexity

• Gconvex is a family of context-specific convexity notions defined by structure-dependent betweenness, ranging from geodesic convexity on manifolds to generalized convex functions.
• In function theory, gconvexity includes methods such as g-convex domination and kernel-defined conjugacy that refine classical convex and Jensen-type inequalities.
• In graph theory, gconvexity refers to closure properties based on shortest paths, leading to diverse computational challenges and applications in optimization.

Searching arXiv for recent and core papers on “gconvex” across its main meanings. gconvex is not a single standardized notion. In the literature considered here, it names several distinct convexity concepts: geodesic convexity of sets and functions on manifolds, $g$-convexity defined by the convexity of $g\circ f$, $g$-convex dominated functions, generalized convex polyhedra and generalized convex functions in infinite-dimensional analysis, $G$-convexity induced by a subset of a Grassmann bundle, and graph-theoretic convexities defined by shortest paths or directed $P_3$-configurations (Mitrea, 2016, Sababheh et al., 2020, Luan et al., 2017, Harvey et al., 2011, Wang et al., 2024, Araújo et al., 23 Jun 2026). The common thread is not a universal definition but the replacement of ordinary Euclidean convexity by a structure-dependent notion of “betweenness,” “support,” or “admissible interpolation.”

1. Terminological scope

The term is therefore best understood as a family resemblance rather than a single invariant definition. In the sources below, the ambient structure determines what “convex” means: geodesics on a manifold, a kernel $\phi(x,y)$ in generalized conjugacy, a comparison function $g$, a class of tangent $p$-planes $G\subset G(p,TX)$, or a path system in a graph.

Context	Core notion	Representative source
Riemannian geometry	Geodesically convex sets; geodesically convex functions along geodesics	(Mitrea, 2016, Wang et al., 2024)
Functional inequalities	$g\circ f$ convex; $g\circ f$0-convex dominated Jensen defect	(Sababheh et al., 2020, Özdemir et al., 2013)
Locally convex TVS and optimization	Generalized convex polyhedra; kernel-defined generalized convex functions	(Luan et al., 2017, Nehzati, 30 Aug 2025)
Geometric potential theory	$g\circ f$1-plurisubharmonicity and $g\circ f$2-convex domains	(Harvey et al., 2011)
Harmonic mapping theory	Convexity in one direction of image domains	(Beig et al., 2017)
Graph theory	Geodetic convexity; $g\circ f$3- and $g\circ f$4-convexities	(Benesh et al., 3 Jun 2026, Araújo et al., 23 Jun 2026)

A persistent misconception is to read gconvex as if it always meant “geodesically convex.” In some papers it does, but in others the letter $g\circ f$5 refers to a comparison function, to “generalized,” or to a fixed geometric datum $g\circ f$6. The semantics are local to the paper’s formalism rather than global across fields.

2. Geodesic convexity on Riemannian manifolds

In the Riemannian setting, $g\circ f$7 most directly means geodesic convexity. For a connected, complete Riemannian manifold $g\circ f$8, a subset $g\circ f$9 is convex in the sense of "Geodesic Convexity Types in Riemannian Manifolds" if, for all $g$0, there exists a unique minimizing geodesic segment $g$1 from $g$2 to $g$3 and $g$4 (Mitrea, 2016). The same paper distinguishes strong convexity, where the endpoints may lie in $g$5 and only the open segment $g$6 must lie in $g$7. This distinction is nontrivial away from Euclidean space because minimizing geodesics need not be unique, and geodesic segments may run along the boundary.

The paper further separates global set-convexity from local sphere conditions. For a geodesic sphere $g$8, the convexity condition requires that every tangent geodesic satisfies $g$9 near the tangency point, while the strong convexity condition strengthens this to strict inequality away from the contact point. These induce the notions of local convexity and strong local convexity of the ball $G$0. The associated pointwise radii

$G$1

measure how far each of these properties persists around $G$2, and they satisfy

$G$3

together with Berger’s global inequality

$G$4

for complete manifolds (Mitrea, 2016).

The central structural question in that work is when several convexity types coincide on all geodesic balls. The paper defines Cpsl by the coincidence of proper convexity, strong convexity, and strong local convexity on balls below the injectivity radius, and Cps by the coincidence of proper and strong convexity on such balls. These hypotheses are highly restrictive. If Cpsl holds at even one point $G$5, then $G$6, and the manifold is diffeomorphic to $G$7. Globally, a complete manifold is Cps if and only if $G$8, and, via O’Sullivan’s result quoted there, this is equivalent to being simply connected and focal-point-free (Mitrea, 2016).

The Euclidean case is the degenerate benchmark: straight lines are the unique minimizing geodesics, all these convexity types collapse, and all corresponding radii are infinite. The point of the paper is precisely that this collapse is exceptional rather than generic.

3. Function-theoretic meanings of $G$9-convexity

For functions on manifolds, geodesic convexity is the direct analogue of ordinary convexity: a function $P_3$0 is geodesically convex if, for every geodesic segment $P_3$1,

$P_3$2

"The sparseness of g-convex functions" identifies this with the positivity of the covariant Hessian,

$P_3$3

on a $P_3$4-convex domain, and then asks a reverse question: given $P_3$5, when does there exist a connection or metric for which $P_3$6 is g-convex? The paper proves that any smooth function with no critical points is g-convex with respect to some connection, that local convexity near every critical point is also sufficient, and that under a geodesically complete connection a g-convex function with discrete critical set has at most one critical point. It then derives three sparseness phenomena: on a compact manifold the set of g-convex smooth functions is nowhere dense in $P_3$7; for generic polynomials on $P_3$8 that are g-convex under some geodesically complete connection, there is at most one critical point; and in several polynomial families the density of g-convex members decreases asymptotically to zero (Wang et al., 2024).

A different functional usage appears in "A new treatment of convex functions." There, $P_3$9 is called $\phi(x,y)$0-convex when $\phi(x,y)$1 is increasing and concave and the composition $\phi(x,y)$2 is convex. This recovers ordinary convexity when $\phi(x,y)$3 and log-convexity when $\phi(x,y)$4. The basic inequality is

$\phi(x,y)$5

which refines Jensen’s inequality. The same paper defines the index of convexity

$\phi(x,y)$6

and shows, for example, that log-convexity implies $\phi(x,y)$7 (Sababheh et al., 2020).

Yet another meaning is $\phi(x,y)$8-convex domination. For a convex $\phi(x,y)$9, a function $g$0 is $g$1-convex dominated if

$g$2

On rectangles $g$3, "On The Coordinated g-convex Dominated Functions" defines coordinate-wise $g$4-convex domination by requiring the corresponding one-variable slices to satisfy this inequality. In one dimension, and coordinate-wise in the paper’s two-variable setting, this is equivalent to convexity of $g$5 and $g$6. The resulting framework yields Hadamard-type and Fejér-type inequalities in which the deviation of $g$7 from the classical convex inequality is controlled by the corresponding deviation of $g$8 (Özdemir et al., 2013).

The Grand Lebesgue Space literature uses gconvex in a weaker geometric sense. "Analog of modulus of convexity for Grand Lebesgue Spaces" does not introduce a full modulus of convexity for GLS; instead it defines a weak characteristic of convexity $g$9 through

$p$0

and specializes this to $p$1. For $p$2 it derives lower bounds $p$3, and for $p$4 it derives $p$5, yielding refined triangle inequalities in GLS. The paper explicitly states that it does not fully prove uniform convexity of all GLS considered and poses the existence of a genuine modulus of convexity for classes such as subgaussian GLS as an open problem (Formica et al., 2021).

4. Generalized convex sets, transforms, and optimization

In infinite-dimensional convex analysis, gconvex frequently means generalized convex rather than geodesic. "A Representation of Generalized Convex Polyhedra and Applications" studies generalized polyhedral convex sets in a locally convex Hausdorff topological vector space $p$6. A subset $p$7 is a generalized convex polyhedron if there exist continuous linear functionals $p$8, scalars $p$9, and a closed affine subspace $G\subset G(p,TX)$0 such that

$G\subset G(p,TX)$1

The core representation theorem states that a nonempty generalized convex polyhedron admits a Minkowski-type representation

$G\subset G(p,TX)$2

with finitely many points $G\subset G(p,TX)$3, finitely many directions $G\subset G(p,TX)$4, and a closed linear subspace $G\subset G(p,TX)$5. The paper then uses this representation to prove Eaves-type and Frank–Wolfe-type existence criteria for generalized linear programming, to characterize the set of linear functionals admitting solutions, and to show that the weakly efficient solution set of a generalized linear vector optimization problem is the union of finitely many generalized polyhedral convex sets (Luan et al., 2017).

A more recent and explicitly computational direction appears in "Universal Representation of Generalized Convex Functions and their Gradients." There the ambient structure is a surplus kernel $G\subset G(p,TX)$6, and the generalized transform of $G\subset G(p,TX)$7 over $G\subset G(p,TX)$8 is

$G\subset G(p,TX)$9

A function is $g\circ f$0-convex if it equals a $g\circ f$1-then-$g\circ f$2 biconjugate, and the paper proves that finitely $g\circ f$3-convex functions

$g\circ f$4

are dense in the full class $g\circ f$5. Under semiconvexity assumptions on $g\circ f$6, the gradients of finitely $g\circ f$7-convex functions are dense in the gradients of all $g\circ f$8-convex functions. The paper further introduces lean parameterizations, proves that the lean parameter set is convex, compares the resulting architecture to shallow neural networks, and uses the Python package gconvex to solve a revenue-maximizing auction problem for multiple goods (Nehzati, 30 Aug 2025).

A plausible implication is that, in this literature, gconvex functions less as a philosophical generalization of convexity than as a parameterized solution class: if the optimizer is known a priori to be generalized convex, one can optimize directly over that class rather than over unrestricted function spaces.

5. Structure-dependent geometric variants

Upper-case $g\circ f$9-convexity is a separate construction. "Geometric plurisubharmonicity and convexity - an introduction" starts from a closed subset $g\circ f$00 of the Grassmann bundle of tangent $g\circ f$01-planes of a Riemannian manifold $g\circ f$02. A smooth function $g\circ f$03 is $g\circ f$04-plurisubharmonic if

$g\circ f$05

The associated $g\circ f$06-convex hull of a set $g\circ f$07 is

$g\circ f$08

and $g\circ f$09 is $g\circ f$10-convex if every compact $g\circ f$11 has $g\circ f$12, equivalently if $g\circ f$13 admits a smooth $g\circ f$14-plurisubharmonic proper exhaustion. The paper also defines strict $g\circ f$15-convexity via a strictly $g\circ f$16-plurisubharmonic exhaustion and proves existence and uniqueness for the Dirichlet problem for $g\circ f$17-harmonic functions on domains with smooth strictly $g\circ f$18-convex boundary and empty $g\circ f$19-core (Harvey et al., 2011). This is formally analogous to the passage from convexity in $g\circ f$20 to pseudoconvexity in several complex variables.

In harmonic mapping theory, the relevant object is usually convexity in one direction rather than full convexity. "Convexity in one direction of convolutions and linear combination of harmonic functions" studies harmonic maps $g\circ f$21 on the unit disk and calls a domain convex in direction $g\circ f$22 when every line parallel to $g\circ f$23 intersects it in an empty set or a line segment. Using the Clunie–Sheil-Small shear construction, the paper proves that specific convolutions of slanted half-plane mappings are convex in the direction $g\circ f$24, that certain right-half-plane convolutions are convex in the direction of the real axis, and that suitable convex combinations of a family $g\circ f$25 remain convex in the direction of the imaginary axis (Beig et al., 2017). In this usage, gconvex is geometric but not geodesic: the operative structure is a preferred family of parallel lines in $g\circ f$26.

A different geometric usage appears in time-frequency analysis. "Gabor orthogonal bases and convexity" does not define a new convexity notion, but it studies Gabor systems with window $g\circ f$27, where $g\circ f$28 is a bounded convex body, symmetric about the origin, with smooth boundary and everywhere non-vanishing Gaussian curvature. The main theorem states that if $g\circ f$29 and $g\circ f$30, then no set $g\circ f$31 yields a Gabor orthonormal basis

$g\circ f$32

for $g\circ f$33 (Iosevich et al., 2017). Here convexity enters through stationary-phase asymptotics for $g\circ f$34, not through a formal definition of gconvex; the term belongs to an overlapping lexical neighborhood rather than to the core definitional family.

6. Graph-theoretic convexities and algorithmic complexity

In graphs, gconvex often means geodetic convexity. "Impartial geodetic removing games on graphs" defines a vertex set $g\circ f$35 to be geodetically convex if it contains every vertex on any shortest path between two vertices of $g\circ f$36. Its convex hull is

$g\circ f$37

which the paper emphasizes is a genuine closure operator, unlike the older geodetic closure. A set is generating if $g\circ f$38, and the paper studies two impartial removing games in which players select vertices until the convex hull of the jointly unselected vertices ceases to be the whole vertex set. The achievement game $g\circ f$39 and the avoidance game $g\circ f$40 are analyzed via maximal nonterminating and minimal terminating sets, leading to explicit nim-values for cycles, hypercubes, complete multipartite graphs, wheel graphs, generalized wheel graphs, and graphs with a unique minimal generating set (Benesh et al., 3 Jun 2026).

A second graph-theoretic meaning is based on directed paths of length two. "Convex geometries and directed paths on three vertices" defines $g\circ f$41-convexity on an oriented graph $g\circ f$42 by requiring that no vertex outside $g\circ f$43 be the central vertex of a directed path $g\circ f$44 with endpoints $g\circ f$45. The $g\circ f$46-variant further requires $g\circ f$47, so the path is induced or shortest. The paper characterizes when these convexities are convex geometries, meaning that every convex set equals the convex hull of its extreme vertices. For $g\circ f$48-convexity, recognition is polynomial-time via a structural characterization involving acyclicity, a distance bound for descendants, and a local $g\circ f$49 condition. For $g\circ f$50-convexity, the situation is markedly harder: deciding whether the convexity is geometric is coNP-complete, although the problem becomes polynomial-time on the class of acyclic indifference oriented graphs (Araújo et al., 23 Jun 2026).

The contrast between these two graph papers illustrates a broader point. Even when gconvex unmistakably refers to a closure operator on subsets, the operative notion of “between” can be shortest paths, induced directed $g\circ f$51’s, or other path systems, and the resulting convexity theory can range from easily recognizable to coNP-complete.

Taken together, these usages show that gconvex is best treated as a contextual technical label. In Riemannian geometry it is about geodesics and covariant Hessians; in analysis it may refer to a transform $g\circ f$52, domination by a convex gauge $g\circ f$53, or kernel-defined generalized conjugacy; in geometric PDE it is tied to a chosen subset $g\circ f$54 of tangent $g\circ f$55-planes; and in graph theory it is a closure system determined by path families. The term is therefore unified by the replacement of Euclidean linear segments with a problem-specific notion of admissible interpolation, but not by a single canonical definition.