Local Convex Envelope Lemma

• The Local Convex Envelope Lemma defines how the convex envelope of a function is determined strictly by its local behavior and supporting hyperplanes.
• It underpins methodologies in PDEs, optimization, and discrete structures by enabling local convexification and regularity via uniform convexity assumptions.
• The lemma facilitates numerical schemes and error estimates by ensuring that under local and uniform convexity, the envelope inherits smoothness and affine behavior on contact sets.

The Local Convex Envelope Lemma is a foundational principle in convex analysis, partial differential equations, functional analysis, and optimization. It asserts that under precise conditions, the value of the convex envelope of a function at a point is determined locally—often by convex combinations of values of the underlying function in a neighborhood, or by the local behavior of the function with respect to supporting hyperplanes or subgradients. This lemma underlies envelope regularity, convexification procedures, and maximum principles, and is expressed in both analytic and geometric terms. The concept has broad applicability, from PDE characterizations (e.g., viscosity solutions for the convex envelope operator) to discrete geometric settings, functional optimization, and nonlocal generalizations.

1. Geometric and Analytic Definitions

The convex envelope of a function $f$ on domain $\Omega$ is the pointwise supremum of all convex functions majorized by $f$: $$\breve{f}(x) = \sup\{ l(x) : l \text{ affine},\ l \leq f \text{ on } \Omega \}.$$ Alternatively, globally and locally: $$\breve{f}(x) = \inf\left\{ \sum_{k=1}^n \lambda_k f(x_k) : x = \sum_{k=1}^n \lambda_k x_k,\ \sum \lambda_k = 1,\ \lambda_k \geq 0,\ x_k \in \Omega \right\},$$ with local versions restricting to $x_k$ close to $x$. For $C^2$ functions, local convexity is equivalent to the Hessian being positive semidefinite at each point, and strict local convexity corresponds to positive definiteness.

In PDE form, the convex envelope solves

$$\inf_{|z|=1} D^2 u(x)[z,z] = 0, \quad u|_{\partial\Omega} = g,$$

i.e., the minimum second directional derivative vanishes in the domain; this is the analytic content of the Local Convex Envelope Lemma in the viscosity sense (Silvestre et al., 2010).

2. Locality Principles and Lemma Statements

A central property is that $\breve{f}(x)$ depends only on the values of $f$ in a neighborhood of $x$, particularly for functions exhibiting discrete or local uniform convexity. For any $\varepsilon > 0$, if $f$ is $\varepsilon$-uniformly convex, then: $$\text{Proposition:}\quad \breve{f}(x) \text{ depends only on } f(x_k) \text{ for } \|x-x_k\| < \varepsilon,$$ allowing convexification to be effected via local supporting averages (Grelier et al., 2021).

In the context of PDEs, the local contact set at $x$,

$C^x = \{y : u(y) = P^x(y)\},$

with $P^x$ a supporting hyperplane at $x$, is convex, contains a segment to the boundary, and governs local flatness—a direction along which $u$ is affine (Silvestre et al., 2010).

On graphs and combinatorial structures (e.g., trees), analogous local rules hold: convexity at a node is characterized by comparison inequalities relating its value to successor values, yielding inductive or recursive computational methods for the envelope (Pezzo et al., 2019).

3. Quantitative Characterizations and Modulus Transfer

For functions with quantitative convexity (uniform convexity), convexification preserves a strengthened version: $$\forall\, \varepsilon' > \varepsilon,\ \exists\, \delta' > 0 \text{ such that } \|x-y\| \geq \varepsilon' \implies \breve{f}\left( \frac{x+y}{2} \right ) \leq \frac{\breve{f}(x) + \breve{f}(y)}{2} - \delta'.$$ That is, the convex envelope inherits discrete uniform convexity of $f$ with mildly weakened separation and modulus (Grelier et al., 2021).

In numerical contexts, local flatness manifests as affine behavior (zero second difference) in specific directions away from the contact set (Li et al., 2018): $u(x \pm \delta v) = u(x) \pm \delta (p \cdot v).$ This feature is crucial for finite-difference schemes and error estimates.

4. Representation Formulas and Duality

In infinite-dimensional or abstract settings, the envelope is represented as

$$f^{**}(x) = \liminf_{y \to x} \inf \left\{ \int f \, d\mu\ :\ \int w\, d\mu(w) = y \right\},$$

where the infimum is over probability measures $\mu$ with prescribed mean (Ruf et al., 2021). For spaces where finitely supported measures suffice (e.g., finite-dimensional), this recovers Carathéodory-type results. Convex envelope regularity (e.g., $C^{1,1}$ smoothness) is assured when $f$ is locally semi-concave and coercive (Fathi et al., 2010).

Dual principles relate convexity to properties of minimizers of tilted functionals: $$f \text{ convex } \iff\ \forall \lambda,\, \text{ set of minimizers of } f(x) - \langle \lambda, x \rangle \text{ is convex.}$$ For strict convexity, uniqueness of minimizers is required (Yu, 2021).

5. Lemma Applications: PDEs, Optimization, and Discrete Structures

The local convex envelope lemma and its consequences inform:

• Maximum principles and envelope estimates for PDEs, e.g., solutions of Monge–Ampère equations have gradients constrained by the convex hull of boundary gradients (Ricceri, 2015).
• Regularization and insertion results, e.g., construction of $C^{1,1}$ interpolants sandwiched between semi-convex and semi-concave functions (Fathi et al., 2010).
• Convexification schemes in nonlinear discrete optimization, e.g., convex envelopes for ray-concave functions with explicit closed-form formulas under facet convexity and positive homogeneity (Barrera et al., 2021).
• Nonlocal generalizations via the “fractional convex envelope”, which as $s \to 1$ (fractional order) converges to the classical envelope and is characterized by vanishing nonlocal directional Laplacians (Pezzo et al., 2020, Barrios et al., 11 Apr 2024).
• Extensions on non-convex domains and adaptation to discrete structures (e.g., trees), where envelopes satisfy natural recursive difference equations (Pezzo et al., 2019, Yan, 2012).

6. Table: Local Convex Envelope Lemma Manifestations

Setting	Local Envelope Dependence	Regularity/Characterization
Banach space, uniform convexity	Values of $f$ in $\varepsilon$-neighborhood	Envelope inherits (nearby) uniform convexity
Viscosity solution (PDE)	Supporting hyperplanes/subdifferentials at $x$	Affine on contact set (zero second derivative in some direction)
Tree/Graph	Node and adjacent/descendant values	Satisfies nonlinear difference equation
Infinite-dimensional vector space	Probability measure moments near $x$	Envelope matches with averages over measures
Finite difference (numerical)	Grid points within scale $h$, $\delta$	Local flatness leads to sharp error bounds

7. Context, Limitations, and Extensions

The local convex envelope lemma does not hold without appropriate convexity or semi-concavity assumptions, coercivity, or domain regularity. For example, absence of coercivity may lead to failure of differentiability of the envelope (Fathi et al., 2010). Nonlocal or fractional settings require continuity and boundedness of exterior data, and strict convexity of the domain for boundary regularity and uniqueness (Pezzo et al., 2020, Barrios et al., 11 Apr 2024). The lemma extends—and often generalizes—classical maximum principles, convex hull properties, and radial representation theories, such as those involving radial uniform upper semicontinuity for extensions to nonconvex, star-shaped sets (Hafsa et al., 2012).

8. Principal References

• Oberman & Silvestre: PDE characterization and regularity of the convex envelope (Silvestre et al., 2010)
• Ricceri: Convex hull properties and envelope dichotomy (Ricceri, 2015)
• Del Pezzo et al.: Convex envelopes on trees (Pezzo et al., 2019)
• Anza Hafsa & Mandallena: Radial representation theorem (Hafsa et al., 2012)
• Barrios et al.: Fractional envelope convergence (Barrios et al., 11 Apr 2024)
• Fathi & Zavidovique: Ilmanen’s lemma and regularity via envelope methods (Fathi et al., 2010)

The Local Convex Envelope Lemma is thus a unifying principle across various domains, governing the local behavior, regularity, and computation of convexifications, and enabling analytic, geometric, and practical advances in convex analysis and related fields.