ε-𝒱𝒰 spaces and approximate leaves\\ for sublinear composite functions
Abstract
Let h be proper, closed and sublinear, and let F be of class C1. The exact 𝒱𝒰 calculus of the composite f=h∘ F is taken as known. This paper studies enlargements ∂̅ε h⊂∂ε h and the resulting ε-spaces of f. Nested inclusions give one-sided Painlevé--Kuratowski limits of 𝒱ε and 𝒰ε at points of the lineality space 𝒰h(0). An additional transversality transfers those limits through F; without it only the exact chain rule is available. Approximate leaves ℳε(x)=F−1(F(x)+𝒰ε h(F(x))) are inner semicontinuous under the nesting 𝒰h(F(x̅))⊂𝒰ε h(F(x)). For a polyhedral outer function the activity set freezes along the exact leaf for small ε, so the dimension identity holds on that leaf and not necessarily in a full neighbourhood of (x̅,0). Partial smoothness of h relative to 𝒰h(0) extends a known result from the origin to the lineality space; it is not used by the transfer theorems.
Related
- Boundary Vanishing Orders and Asymptotic Expansions for Planar Harmonic Functions with a Sublinear Robin Condition
- Small-time approximate controllability of bilinear Schrödinger equations on compact normal homogeneous spaces
- Approximate Controllability of Neutral Integro-Differential Inclusions Associated with Hemivariational Inequalities in Banach Spaces with Uniformly Convex Dual
- Norming Approximate Orthogonality in Normed Linear Spaces
- On the Minimum Number of Linear Pieces Required to Approximate Nonlinear Functions under an Accuracy Constraint
BibTeX
@misc{liu2026ε𝒱𝒰,
title = {{$\varepsilon$-$\mathcalVU$ spaces and approximate leaves\\ for sublinear composite functions}},
author = {Shuai Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25083},
eprint = {2610.25083},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25083},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - ε-𝒱𝒰 spaces and approximate leaves\\ for sublinear composite functions
AU - Shuai Liu
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.25083
UR - https://arxiv.org/abs/2610.25083
AB - Let h be proper, closed and sublinear, and let F be of class C^1. The exact 𝒱𝒰 calculus of the composite f=h∘ F is taken as known. This paper studies enlargements ∂̅_ε h⊂∂_ε h and the resulting ε-spaces of f. Nested inclusions give one-sided Painlevé--Kuratowski limits of 𝒱_ε and 𝒰_ε at points of the lineality space 𝒰h(0). An additional transversality transfers those limits through F; without it only the exact chain rule is available. Approximate leaves ℳ_ε(x)=F^−1(F(x)+𝒰_ε h(F(x))) are inner semicontinuous under the nesting 𝒰h(F(x̅))⊂𝒰_ε h(F(x)). For a polyhedral outer function the activity set freezes along the exact leaf for small ε, so the dimension identity holds on that leaf and not necessarily in a full neighbourhood of (x̅,0). Partial smoothness of h relative to 𝒰h(0) extends a known result from the origin to the lineality space; it is not used by the transfer theorems.
ER -