Supporting Sets in Dual Spaces

• Supporting sets in dual spaces are subsets that capture the dual support of convex, projective, and algebraic structures, enabling a translation of primal problems into dual formulations.
• They utilize mechanisms such as the Galois connection and polar calculus to reverse operations like intersection and summation, simplifying geometric and optimization challenges.
• These concepts find applications in variational analysis, Banach space theory, and algebraic geometry, providing actionable insights for computational methods and theoretical classification.

Supporting sets in dual spaces encompass a range of advanced concepts in functional analysis, convex geometry, and duality theory. In general, a supporting set refers to a subset of a dual space that encodes how the primal structure (e.g., convex sets, cones, or algebraic ideals) is “supported” or “separated” by dual objects such as hyperplanes, functionals, or other dual-specific constructs. Recent research has provided a diverse array of perspectives, from deep duality frameworks in convex and projective geometry to the role of supporting sets in algorithmic and geometric analysis.

1. Duality and Supporting Sets in Projective and Convex Geometry

Saturated multi-convex sets in real projective spaces exemplify supporting sets in dual spaces through a categorical duality. For a real projective space $\mathbb{P}$ and its dual $\mathbb{P}^*$, every nonempty saturated multi-convex set $S \subsetneq \mathbb{P}$ can be reconstructed as the intersection of all irreducible convex sets (complements of hyperplanes) containing $S$. These irreducible convex sets are in bijection with points of the dual space. The key duality is formalized as a Galois connection:

$$\Phi(S) = \{\delta(C) \in \mathbb{P}^* : S \subset C,\; C\ \text{irreducible}\}$$

where $\delta$ is the correspondence between irreducible convex sets in $\mathbb{P}$ and points in $\mathbb{P}^*$. This establishes an order anti-isomorphism:

$S \subset T \iff \Phi(T) \subset \Phi(S)$

with $\Phi(\Phi(S)) = S$ for saturated multi-convex sets. In practical terms, this means the family of supporting hyperplanes to a convex or multi-convex set is precisely encoded via a dual subset in $\mathbb{P}^*$; computational and combinatorial properties (such as those used in discrete geometry for transversal and separation theorems) are thus translated via duality to typically simpler or more tractable problems (1005.1852).

2. Geometric Duality of Cones and Polar Calculus

In dual pairs of vector spaces $(Y, Z)$, supporting sets arise as polars of convex sets and cones, concretely defined as

$$A^\circ = \{z \in Z : \langle y, z \rangle \le 1\ \forall y \in A\}\,,$$

with similar definition for subsets of $Z$. The relationships between sets in $Y$—such as combinations of cones $C$, $D$ and convex sets $B_1$, $B_2$—and their polars in $Z$, precisely capture the dual interplay between geometric properties (normality, conormality, additivity, coadditivity). For example, normality in $Y$:

$(B_2 + C) \cap D \subseteq B_1$

is dual to conormality in $Z$:

$B_1^\circ = B_2^\circ + C^\circ$

provided suitable closedness hypotheses. Polars act as supporting sets that reflect and reverse the operations of sum and intersection between cones and convex sets, thus "dualizing" the primal geometric structure. This framework extends to practical applications in C*-algebra contexts, where norms and decompositions are described via such duality among supporting cones and convex subsets (Messerschmidt, 2014).

3. Supporting Sets and Dual Formulations in Variational Analysis

Supporting sets in dual spaces are crucial in dual characterizations of regularity and separation properties for collections of sets in Banach and Hilbert spaces. In subtransversality and intrinsic transversality theory, the normal cone construction in the dual provides supporting sets witnessing geometric properties such as error bounds and regular intersections. For closed sets $A, B$, and for points $a \in A$, $b \in B$, dual vectors $x_1^* \in N_A(a)$ and $x_2^* \in N_B(b)$ form a supporting set if, after suitable normalization,

$$\|x_1^* + x_2^*\| > \alpha,\quad \|x_1^*\| + \|x_2^*\| = 1$$

holds. This condition ensures that the supporting normals do not "cancel," underpinning geometric regularity (e.g., subtransversality or intrinsic transversality) and linear convergence guarantees for projection algorithms (Kruger et al., 2016, Thao et al., 2018, Bui et al., 2019). For collections of sets, systems of dual vectors $\{x_i^*\}$ provide generalized separation—supporting the system collectively by satisfying normalization and sum-to-zero conditions, thereby acting as a "dual certificate" of extremality or stationarity.

4. Order-Theoretic and Topological Perspectives: Support, Stone Duality, and Hochster Duality

Supporting sets acquire an order-theoretic and topological context in the analysis of Stone-type dualities for lattices and categories. The support of an element $a$ in a lattice $L$ is defined as

$$\operatorname{supp}(a) = \{ I \in \operatorname{Spc}(L) : a \notin I \}$$

where $\operatorname{Spc}(L)$ denotes the set of prime ideals, topologized appropriately. The assignment $a \mapsto \operatorname{supp}(a)$ transfers lattice-theoretic information to topological (spectral) spaces. Such supports provide the closed sets in the Stone spectrum, while in the Hochster dual topology, the same supports become basic open sets. This dual perspective is essential for classifying thick tensor ideals in tensor exact categories, representation categories, and tensor triangulated geometry (Krause, 2023).

5. Supporting Sets in Duals of Banach and Function Spaces

In the setting of Banach spaces and function spaces, supporting sets in dual spaces appear as weak*-closures, derived sets, or strong subdifferentiability points.

• Weak*-derived sets and their transfinite iterations: For a subset $A$ of a dual $X^*$, the hierarchy $A^{(\alpha)}$ of weak*-derived sets (obtained by iteratively collecting weak*-limits of bounded nets) encodes layers of support properties. In non-reflexive Banach spaces, convex sets can be found whose support via weak*-limits requires arbitrarily high countable ordinal iterations to stabilize, exhibiting a much more complex dual support structure than in the reflexive case (Silber, 2021, Ostrovskii, 2021).
• Strongly subdifferentiable points: In dual $L^1$-spaces and related Banach spaces, points of strong subdifferentiability (SSD-points) are functionals whose kernels are strongly proximinal in the predual. SSD-points represent "supporting" elements as they attain their norm and their kernels correspond to best-approximation subspaces, forming a discrete part of the overall dual structure (Jayanarayanan et al., 2020).
• Closedness and dual representation in function spaces: In Orlicz spaces, the coincidence between order closedness and weak-closedness (under the Krein–Smulian property) ensures dual representations for convex risk measures, where the supporting sets are acceptance sets in the dual. Breakdown of this equivalence leads to failure of dual representation for certain coherent risk measures (Gao et al., 2016).

6. Combinatorial and Algorithmic Constructions of Supporting Sets

Recent work provides combinatorial and algorithmic frameworks for supporting sets in duals:

• Supporting sets via quasi-Banach duals: For combinatorial Banach spaces defined by hereditary families, the quasi-norm

$$\|x\|^{\mathcal{F}} = \inf_{\mathcal{P} \in \mathbb{P}_{\mathcal{F}}} \sum_{F \in \mathcal{P}} \sup_{k \in F} |x(k)|$$

on $c_{00}$ (the finitely supported sequences) provides a quasi-Banach version of the dual. The geometric properties of these spaces—such as $\ell_1$-saturation and absence of the Schur property for large families—are determined by the structure of supporting sets formed by extreme points corresponding to maximal combinatorial subsets (Borodulin-Nadzieja et al., 2022).

• Supporting sets in polynomial and Macaulay dual spaces: In computational invariant theory and algebraic geometry, supporting sets are realized as multi-graded components of the Macaulay dual space of a polynomial ideal. Algorithms are provided for recursively constructing these graded supporting sets via lattice point orderings and closedness operators, enabling effective computation of bases and dual representations for saturated and quotient ideals (Cummings et al., 2023).

7. Transformations, Dual Curves, and Envelopes in Classical Geometry

Supporting sets in dual spaces also have classical manifestations in differential and projective geometry. The family of tangent or supporting lines to a curve $y = y(x)$ is mapped to a dual curve in the space of line coefficients (e.g., slope and intercept). The Legendre transform

$d(m) = mx - y$

relates points and tangents, with the set of all such $(m, d(m))$ forming the dual curve. This construction provides a reflexive duality between the original geometric set and its aggregate of supporting lines, encapsulating geometric properties—such as convexity, inflection, and envelopes—through dual equations and transformations (Kilner et al., 2021).

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Overall, supporting sets in dual spaces are central elements in the duality theory of convex, topological, algebraic, and combinatorial structures. They provide a precise way to describe, encode, and manipulate the “support” of a set, whether it is via hyperplanes in geometric spaces, polar sets in dual vector spaces, supports of elements in spectral spaces, or combinatorial partitions in sequence spaces. These tools yield both deep theoretical insight and robust practical approaches for computation, optimization, and classification across multiple mathematical disciplines.