General Separation in ℝⁿ

• General Separation in ℝⁿ is a dual theorem defining clear conditions to separate multiple sets using variational principles and product norms.
• It establishes balance and normalization conditions that ensure separation even for nonconvex and multi-set configurations.
• The approach employs Ekeland’s principle and subdifferential calculus, providing insights for optimality, approximate stationarity, and transversality.

A general separation result in ℝⁿ refers to a dual or variational theorem that provides necessary (and sometimes sufficient) conditions for the mutual non-intersection, controlled overlap, or extremality properties of multiple sets—often going far beyond classical hyperplane separation for two convex sets. Contemporary developments unify and extend these statements using product norms and variational methods, accommodating nonconvex sets, collections of more than two sets, and yielding dual characterizations central to variational analysis and optimization.

1. Conceptual Framework and Scope

A generalized separation result considers a finite collection of sets $\Omega_1, \ldots, \Omega_n$ in a normed linear space $X$ (with $X = \mathbb{R}^n$ as a canonical example), aiming to state dual conditions under which these sets can be "separated" in a suitable sense. Instead of the existence of a hyperplane strictly separating two convex sets, generalized separation statements typically:

• Allow nonconvexity of the sets.
• Address collections with $n \geq 2$.
• Express separation through dual objects (e.g., elements of the dual space $X^*$, normal cones, subdifferentials).
• Utilize arbitrary product norms on $X^{n-1}$ or $X^n$ to control the dual conditions.

Suppose the sets $\Omega_1, \dots, \Omega_n$ are so positioned that their intersection is empty in a robust way (e.g., even under small translations), the result asserts the existence of nontrivial $(x_1^*, \dots, x_n^*) \in (X^*)^n$ satisfying a balance constraint and specific normalization. These dual elements witness the separation of the family.

2. Role of Product Norms and Compatibility

Central to the general separation result is the explicit use of product norms on $X^{n-1}$ (or $X^n$). Let $\| \cdot \|$ denote the base norm on $X$, and define a product norm, e.g.,

$$\|(u_1, \ldots, u_{n-1})\| := \left( \sum_{i=1}^{n-1} \|u_i\|^p \right)^{1/p} \quad \text{or} \quad \|(u_1, \ldots, u_{n-1})\|_{\max} := \max_{1\le i\le n-1} \|u_i\|$$

The proof and statement rely on compatibility conditions (labeled (C1)–(C6) in (Cuong et al., 6 Dec 2024)) between the product norm and the base norm, which guarantee that the dual normalization and balance constraints are well-posed when moving between $X$ and the product spaces.

The flexibility of the product norm enables the theorem to recover numerous special cases from the literature, such as p-weighted separation statements and the case for the supremum norm, by substituting into the general structure.

3. Statement and Structure of the Separation Result

Let $\Omega_1, \ldots, \Omega_n$ be subsets of $X$ such that no $n$-tuple $$(\omega_1, \ldots, \omega_n) \in \Omega_1 \times \dots \times \Omega_n$$ has $(\omega_1, \ldots, \omega_{n-1}) = \omega_n$ in the product norm ball around some fixed point. If this robust non-intersection holds, then for each such configuration, there exist dual vectors $(x_1^*, \ldots, x_n^*) \in (X^*)^n$, not all zero, with

$$\sum_{i=1}^n x_i^* = 0 \qquad \text{and} \qquad \|(x_1^*, \ldots, x_{n-1}^*)\| = 1,$$

as well as the dual–primal pairing property

$$\sum_{i=1}^{n-1} \langle x_i^*, x_n - x_i \rangle = \|(x_n - x_1, \dots, x_n - x_{n-1})\|.$$

Moreover, there are proximity conditions relating $x^* := (x_1^*, \ldots, x_n^*)$ to the normal cones $N_{\Omega_i}(x_i)$ or their Fréchet/Clarke variants, potentially in a "fuzzy" fashion (that is, up to a small error $\delta$ in the dual norm): $d\big(x^*, N^F_{\widehat\Omega}(x)\big) < \delta$ where $N^F_{\widehat\Omega}(x)$ denotes the product normal cone at $x = (x_1, ..., x_n)$.

Thus, separation is not given in terms of a linear functional or hyperplane, but through the existence of dual certificates satisfying a set of algebraic, normalization, and proximity relations.

4. Proof Strategy and Variational Ingredients

The proof in (Cuong et al., 6 Dec 2024) proceeds via a variational approach:

• Variational Problem: An auxiliary function $$f_1(x_1, ..., x_n) = \|(x_1 - x_n, ..., x_{n-1} - x_n)\|$$ is minimized over the product set $\Omega_1 \times \cdots \times \Omega_n$ or its neighborhood.
• Ekeland’s Principle: The Ekeland Variational Principle is applied to perturb $f_1$ (along with gauge and indicator terms) to obtain an $\varepsilon$-minimizer $x^\circ$ at which subdifferential calculus applies.
• Subdifferential Calculus: Subdifferential sum rules yield the existence of dual elements $(y_1^*, \ldots, y_n^*)$ from the subdifferentials of the component functions at $x^\circ$, which fulfill key balance and normalization conditions.
• Dual Assembly: These duals are then combined, yielding the normalized tuple $(x_1^*, ..., x_n^*)$ with the desired dual balance and normalization, explicitly tied to the product norm.

This modular methodology undergirds the proof's generality and reveals how subdifferential calculus, variational minimization, and product norm structure interact in the separation theorem.

5. Applications: Optimality, Approximate Stationarity, and Transversality

The main applications of the general separation result are in the derivation of dual necessary (and sometimes sufficient) conditions for geometric properties of sets in variational analysis:

• Approximate Stationarity: For sets that almost but not quite intersect, the existence of such dual certificates provides necessary conditions for their "near-extremality" and quantifies the failure of intersection in dual terms.
• Transversality: For collections that intersect in a "regular" way, dual certificates provided by separation theorems can characterize transversality; the balance/normalization conditions have precise interpretations as strong dual regularity properties.
• Nonconvex Optimization: Since the sets may be nonconvex, these dual conditions yield generalized optimality conditions (e.g., multiplier rules) for collections of constraint sets, extending beyond the Polyak extremal principle.
• Unification of Results: By varying the product norm, one recovers as corollaries classical separation theorems, extremal principles, and recent weighted/non-weighted separator statements across the literature.

6. Comparison to Classical and Existing Theorems

The general separation result in (Cuong et al., 6 Dec 2024) unifies and extends several previous frameworks:

• Convex Separation: The classical separation of two convex sets in $\mathbb{R}^n$ by a hyperplane (e.g., Hahn–Banach) is obtained as a special case.
• KKM and Minimax Theorems: The dual structure and normalization conditions recall those underlying the convex KKM principle and the Sion–von Neumann minimax principle, though the context is now nonconvex and multi-set.
• Extremal Principle: The theorem encompasses and generalizes extremal principles, such as those based on the maximum or $p$-norm, and expresses the connection via compatibility of product norms.
• Variational Approaches: The modular proof via Ekeland’s principle and subdifferential sum-rule is new in its scope and systematic adaptability to differing norm structures, illustrating the flexibility of the method for various dual characterizations.

7. Mathematical Highlights and Representative Formulas

Key mathematical formulas central to the theory include:

• Dual Balance and Normalization:

\begin{align*} \sum_{i=1}^n x_i^* &= 0 \ |(x_1^*,..., x_{n-1}^*)| &= 1 \end{align*}

• Primal–dual Pairing:

$$\sum_{i=1}^{n-1} \langle x_i^*, x_n - x_i \rangle = \|(x_n - x_1, \ldots, x_n - x_{n-1})\|$$

• Proximity to Normal Cones:

$d( x^*, N^F_{\widehat\Omega}(x)) < \delta$

These combine dual algebraic requirements, normalization with respect to the product norm, and proximity in the dual space to normal cones associated to each set.

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In summary, a general separation result in $\mathbb{R}^n$ as established in (Cuong et al., 6 Dec 2024) synthesizes convex, nonconvex, multi-set, and multi-norm settings into a unified framework using variational principles on product spaces. This result not only subsumes known separation theorems but unlocks new analysis in dual characterizations of optimality, constraint systems, and regularity phenomena fundamental to variational analysis and optimization.