Extendable Convexity: Theory & Applications

• Extendable convexity is a framework that extends classical convexity properties by using minimal boundary data to construct controlled extensions of functions, sets, and measures.
• The framework employs convex roof constructions, convex-analytic duality, and smooth extension criteria to guarantee continuity and differentiability under precise conditions.
• Its applications span quantum information, optimization, and discrete geometry, offering algorithmic approaches for convex hull characterization and extension decision problems.

Extendable convexity is a unified framework for the extension of convexity properties—whether of functions, sets, or combinatorial structures—beyond classical convex domains, often by specifying sufficient or minimal boundary data and characterizing when and how convex objects (functions, sets, measures) admit continuous, smooth, or otherwise controlled extensions to larger ambient spaces. The modern theory encompasses convex roof constructions, function and set extensions in Euclidean, Hilbert, and more abstract settings, and regularity theory, while also connecting to discrete convex analysis, optimization, quantum information, and algebraic geometry. The guiding principle is to maximize the propagation of convex-like structure from sparse data, through explicit constructions and necessary and sufficient criteria.

1. Convex Roof Extension and Canonical Function Extension

The convex roof (or convex extension) construction provides the canonical procedure for extending a function $f:C \to \mathbb{R}\cup\{+\infty\}$ defined on a compact (not necessarily convex) set $C \subset \mathbb{R}^d$ to its convex hull $K = \mathrm{co}(C)$ by

$$f^*(x) = \inf_{\substack{(t_i, c_i) \ \sum t_i c_i = x,\, c_i\in C,\, t_i\geq 0,\, \sum t_i=1}} \sum t_i f(c_i),$$

where by Carathéodory’s theorem the infimum need only be over combinations of at most $d+1$ points of $C$. This $f^*$ is convex on $K$, agrees with $f$ on $C$, and is the largest convex minorant of all convex extensions of $C \subset \mathbb{R}^d$0 to $C \subset \mathbb{R}^d$1 (Bucicovschi et al., 2010).

Sharp necessary and sufficient conditions for the continuity of $C \subset \mathbb{R}^d$2 are provided: if $C \subset \mathbb{R}^d$3 contains the full relative boundary $C \subset \mathbb{R}^d$4 of its convex hull and $C \subset \mathbb{R}^d$5 is continuous and convex on $C \subset \mathbb{R}^d$6, then $C \subset \mathbb{R}^d$7 is continuous on $C \subset \mathbb{R}^d$8. Conversely, if $C \subset \mathbb{R}^d$9 does not contain the boundary, continuity can fail or no continuous convex extension may exist. Regularity results are also obtained: if $K = \mathrm{co}(C)$0 is $K = \mathrm{co}(C)$1 and $K = \mathrm{co}(C)$2 is $K = \mathrm{co}(C)$3 on the boundary and admits nonvertical supporting hyperplanes everywhere, then $K = \mathrm{co}(C)$4 is globally $K = \mathrm{co}(C)$5 on $K = \mathrm{co}(C)$6.

A celebrated application is in quantum information theory, where for the compact convex set of density matrices, extending an entanglement measure from pure states to mixed states is achieved via the convex roof, guaranteeing continuity of the extension under suitable conditions (Bucicovschi et al., 2010, Uhlmann, 2011).

2. Necessary and Sufficient Conditions for Function and Smooth Convex Extension

For finite data or partial functions $K = \mathrm{co}(C)$7, convex extension to $K = \mathrm{co}(C)$8 is characterized via convex-analytic duality: the extension exists if the polyhedron $K = \mathrm{co}(C)$9 has suitable vertices matching the data, and the canonical extension at $$f^*(x) = \inf_{\substack{(t_i, c_i) \ \sum t_i c_i = x,\, c_i\in C,\, t_i\geq 0,\, \sum t_i=1}} \sum t_i f(c_i),$$0 can be written as $$f^*(x) = \inf_{\substack{(t_i, c_i) \ \sum t_i c_i = x,\, c_i\in C,\, t_i\geq 0,\, \sum t_i=1}} \sum t_i f(c_i),$$1 (Bhaskar et al., 2018). Extension existence can be decided in polynomial time, but evaluation of the extension at arbitrary points is strongly NP-hard unless the ambient dimension is fixed.

In smooth categories, the Whitney extension framework has been adapted to convexity. Given a compact convex $$f^*(x) = \inf_{\substack{(t_i, c_i) \ \sum t_i c_i = x,\, c_i\in C,\, t_i\geq 0,\, \sum t_i=1}} \sum t_i f(c_i),$$2 and a convex function $$f^*(x) = \inf_{\substack{(t_i, c_i) \ \sum t_i c_i = x,\, c_i\in C,\, t_i\geq 0,\, \sum t_i=1}} \sum t_i f(c_i),$$3 with compatible $$f^*(x) = \inf_{\substack{(t_i, c_i) \ \sum t_i c_i = x,\, c_i\in C,\, t_i\geq 0,\, \sum t_i=1}} \sum t_i f(c_i),$$4 Taylor polynomials $$f^*(x) = \inf_{\substack{(t_i, c_i) \ \sum t_i c_i = x,\, c_i\in C,\, t_i\geq 0,\, \sum t_i=1}} \sum t_i f(c_i),$$5 at each $$f^*(x) = \inf_{\substack{(t_i, c_i) \ \sum t_i c_i = x,\, c_i\in C,\, t_i\geq 0,\, \sum t_i=1}} \sum t_i f(c_i),$$6, Azagra-Mudarra established that the existence of a global convex $$f^*(x) = \inf_{\substack{(t_i, c_i) \ \sum t_i c_i = x,\, c_i\in C,\, t_i\geq 0,\, \sum t_i=1}} \sum t_i f(c_i),$$7 (resp. $$f^*(x) = \inf_{\substack{(t_i, c_i) \ \sum t_i c_i = x,\, c_i\in C,\, t_i\geq 0,\, \sum t_i=1}} \sum t_i f(c_i),$$8) extension matching the prescribed jets is equivalent to satisfaction of a hierarchy of convex-Jet inequalities (CW$$f^*(x) = \inf_{\substack{(t_i, c_i) \ \sum t_i c_i = x,\, c_i\in C,\, t_i\geq 0,\, \sum t_i=1}} \sum t_i f(c_i),$$9), which control the lower bounds of directional Taylor expansions to order $d+1$0. In the $d+1$1 case, no loss of smoothness is incurred; for $d+1$2 one typically loses $d+1$3 derivatives unless $d+1$4 has strongly convex $d+1$5 geometry, in which case only one derivative is lost (Azagra et al., 2015).

For $d+1$6 and strongly convex extensions, finiteness principles hold: there exist constants $d+1$7 and $d+1$8 (depending only on dimension) such that if all $d+1$9-point restrictions of the data admit extensions with prescribed strong convexity modulus and Lipschitz bounds, then there exists a global extension with controlled modulus and Lipschitz constant. In one dimension, $C$0 is sharp (Drake, 2024). For general Hilbert spaces, the existence of $C$1 convex extensions is controlled by a quadratic lower bound on divided differences of the data and gradient—(CW1,1)—and sharp interpolation results for the boundaries of convex bodies are obtained (Azagra et al., 2016).

3. Set-Theoretic and Polyhedral Extension: Geometry, Algorithms, and Discrete Convexity

Set-theoretic approaches, rooted in geometric measure theory and combinatorial optimization, provide algorithmic schemes for convex hull characterization and proofs. Zuckerberg's method—generalized via set characterizations—constructs associated measurable sets $C$2 for each coordinate of a point $C$3, whose measure matches $C$4, so that varying $C$5 in $C$6 gives indicator combinations tracking the vertex data or functional constraints. This technique extends from 0/1-polytopes to arbitrary polyhedra, general convex hulls, and unbounded convex cones, and underpins practical algorithms for combinatorial optimization (e.g., flow and matching polytopes) (Bärmann et al., 2021).

For discrete convexity, the theory of L-convexity on oriented modular graphs extends classical and combinatorial convexity—via barycentric subdivision, Boolean-gated sets, and modular or polar semilattices. The Lovász extension to orthoscheme complexes (CAT(0) simplicial spaces) provides the convex analytical bridge, with convexity of the Lovász extension equivalent to submodularity or L-convexity on the combinatorial structures (Hirai, 2016). Persistency and optimality criteria for combinatorial relaxations, as well as steepest-descent algorithms with provable polynomial complexity bounds, follow in this framework.

4. Extensions Across Nonconvex or Thin Domains, Regularity, and Barrier Results

In the general Euclidean or Banach setting, convexity may be defined globally, locally, or via interval (segment) convexity, and these notions do not always coincide off convex domains. The convex roof gives the maximal convex extension under mild boundedness or interiority hypotheses (Yan, 2012).

A crucial result is that a finite convex-valued function on a bounded convex domain $C$7 admits a finite convex extension to $C$8 if and only if $C$9 is Lipschitz; the extension is given by explicit sup-inf formulas. For smoothness, convolution and partition of unity techniques allow gluing and smoothing of convex extensions, preserving Hessian convexity when data are $f^*$0 and the Hessian is positive definite on a neighborhood.

If the original domain is convex except on a "thin" set (low codimension), interval convexity, local convexity, and global convexity coincide on the complement. In contrast, removing a codimension-1 barrier can break global convexity even when interval and local convexity persist (Yan, 2012).

In polyconvexity and nonlinear elasticity, convex extensions from a hyperplane to a half-space are achieved via Moreau–Yosida envelopes, providing unique lower semicontinuous convex extensions smooth on the interior of the half-space and matching the original function at the boundary (Ball et al., 2023). This clarifies minimal requirements in applications such as the modeling of materials with blow-up constraints.

5. Generalizations: Fractional, Interpolated, and Relative Convexity

Extendable convexity has been generalized in several directions:

• Fractional convexity introduces an $f^*$1-convexity notion via nonlocal (fractional Laplacian) segment conditions, with the corresponding $f^*$2-convex envelope characterized as a solution to a nonlocal fully nonlinear PDE. This recovers the classical convex envelope as $f^*$3 and connects to fractional Monge-Ampère equations (Pezzo et al., 2020).
• Parametric interpolation between convexity and quasiconvexity is achieved by varying the $f^*$4 operator between second derivative and square gradient moduli. The $f^*$5-convex envelope interpolates between the convex and quasiconvex envelopes as unique solutions to a family of fully nonlinear PDEs. Regularity up to $f^*$6 is obtained under “not V-shaped” ($f^*$7) conditions on the boundary data (Blanc et al., 2023). Each interpolated envelope admits generalized supporting hyperplanes at every point.
• Relative (local) convexity: Points of convexity are defined relative to neighborhoods where the usual Jensen-type inequalities hold locally even if $f^*$8 is not globally convex. This local extendable convexity supports the persistence of key inequalities (Jensen, majorization, risk aversion) under minimal local convexity assumptions, thus broadening the classical domain of convex inequalities (Niculescu et al., 2014).
• Projective and semialgebraic convexity: In algebraic and combinatorial geometry, extendable convexity encompasses subsets of embedded projective varieties—for example, the amplituhedron in the Grassmannian. Here, the set is said to be extendably convex if it equals the intersection of the ambient variety with a convex polytope. This property is verified for the amplituhedron in the $f^*$9 case and relates to properties such as duality, parity, and the structure of the exterior cyclic polytope (Mazzucchelli et al., 23 Jul 2025).

6. Applications and Algorithmic Consequences

The extendable convexity framework yields actionable criteria and algorithms in various fields:

• Quantum information: Construction of continuous convex extensions for entanglement measures using the convex roof and exploitation of symmetries for analytic solutions (Bucicovschi et al., 2010, Uhlmann, 2011).
• Learning and property testing: Efficient algorithms for deciding extendibility of convex and submodular function data; property testing and hardness of evaluating canonical extensions (Bhaskar et al., 2018).
• Optimization: Construction of smooth convex bodies with desired boundary data and normals; interpolation theorems for convex $K$0 and $K$1 extensions (Drake, 2024, Azagra et al., 2016, Azagra et al., 2015).
• Combinatorial optimization: Algorithmic convex-hull proofs and polyhedral characterization via set characterizations and measure-based constructions (Bärmann et al., 2021).
• Nonconvex analysis and duality theory: Extension of the Polyak convexity principle guarantees the persistence of uniform convexity under certain nonlinear maps, supporting the analysis of solution existence, uniqueness, duality, and optimality in constrained nonconvex problems (Uderzo, 2017).

Extendable convexity thus forms a foundational and unifying principle bridging convex analysis, geometry, optimization, discrete mathematics, and applied fields where controlled convex extension and relaxation are essential.