Overview
This research introduces the concept of a directionally fine family of cones, characterizing it as a local geometric property. This property describes the capacity to select cones of arbitrary narrowness around any given direction within a normed space. The framework establishes that this local geometric attribute serves as a common mechanism for both finite separation and finite approximation processes.
Research Context
The study investigates the implications of directionally fine families of cones for mathematical operations such as separation and approximation. It posits that this property provides a unifying mechanism for these operations. The work contrasts with traditional requirements for convexity by demonstrating that the principles developed do not necessitate convexity for the cones being separated. The utility of these constructions extends to generating max-type positively homogeneous separators.
Approach
The research establishes directionally fine families of cones as a local geometric property. It then explores its applications in two primary areas: finite separation and approximation. For separation, the mechanism is applied to directionally compact sets, separating them from closed cones using a finite number of family members. For approximation, it applies to locally compact cones, yielding finite outer approximations. These approximations come with quantitative Hausdorff control over their sets of directions and their bounded sections.
Subsequently, the study delves into two distinct realizations of this framework. The first involves the classical family of Bishop–Phelps cones. Here, directional fineness is characterized by specific geometric conditions: every point on the unit sphere must be a denting point of the unit ball. This condition is equivalent to property $(G)$. In the context of Banach spaces, this translates to the combination of rotundity and the Kadec property.
The second realization introduces a novel construction principle termed transversal coercivity. This principle is presented as a general axial construction method capable of yielding directionally fine families in any normed space, without requiring specific geometric assumptions on the norm. Canonical models arising from transversal coercivity include uniform axial deviation and norm-normalized axial cones. The explicit structural characteristics derived from the latter realization (transversal coercivity) are then leveraged to provide penalization and recovery results relevant to optimization problems involving locally compact cones, alongside quantitative approximation and convergence estimates.
Findings
- Directionally fine families of cones are defined as a local geometric property enabling the selection of arbitrarily narrow cones around every direction in a normed space.
- This property provides a common mechanism for finite separation: directionally compact sets can be separated from closed cones by finitely many members of the family.
- The property also facilitates finite outer approximations for locally compact cones, offering quantitative Hausdorff control over their sets of directions and bounded sections.
- These finite constructions naturally lead to max-type positively homogeneous separators.
- The constructions do not require convexity of the cones for separation.
- For Bishop–Phelps cones, directional fineness is characterized by every point of the unit sphere being a denting point of the unit ball, equivalent to property $(G)$.
- In Banach spaces, this characterization for Bishop–Phelps cones corresponds to rotundity together with the Kadec property.
- Transversal coercivity is introduced as a general axial construction principle.
- Transversal coercivity yields directionally fine families in arbitrary normed spaces without requiring geometric assumptions on the norm.
- Uniform axial deviation and norm-normalized axial cones are identified as canonical models generated by transversal coercivity.
- The explicit structure derived from transversal coercivity results in penalization and recovery results for optimization over locally compact cones, alongside quantitative approximation and convergence estimates.
Why This Matters
The development of directionally fine families of cones offers a unified mechanism for finite separation and approximation, extending applicability beyond convex cones. This framework informs the construction of max-type positively homogeneous separators, potentially impacting optimization strategies. The findings regarding Bishop-Phelps cones and the introduction of transversal coercivity provide new theoretical tools for analyzing geometric properties in normed spaces.
Potential Applications
The explicit structure derived from the transversal coercivity construction principle yields penalization and recovery results for optimization over locally compact cones. This framework also provides quantitative approximation and convergence estimates relevant to these optimization processes.