Borel selection of dominating hyperplanes
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866910061347995648 |
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| author | Clerico, Eugenio |
| author_facet | Clerico, Eugenio |
| contents | We study a natural measurable selection problem for which the standard uniformisation theorems do not seem to apply directly, yet a Borel selector exists. More precisely, we consider families of finite dimensional functions that admit pointwise domination by affine functionals and ask whether such dominating functionals can be chosen in a Borel measurable way. We prove that this is indeed possible under semi-analytic regularity assumptions. The proof combines a one-dimensional Borel insertion result between an upper and a lower semi-analytic functions, derived from Lusin's separation theorem, with an induction on the dimension. As an application, we obtain Borel measurable selections of subgradients for parameter-dependent finite-dimensional convex functions, outside the scope of the standard normal integral framework. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_19973 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Borel selection of dominating hyperplanes Clerico, Eugenio Logic Functional Analysis Optimization and Control We study a natural measurable selection problem for which the standard uniformisation theorems do not seem to apply directly, yet a Borel selector exists. More precisely, we consider families of finite dimensional functions that admit pointwise domination by affine functionals and ask whether such dominating functionals can be chosen in a Borel measurable way. We prove that this is indeed possible under semi-analytic regularity assumptions. The proof combines a one-dimensional Borel insertion result between an upper and a lower semi-analytic functions, derived from Lusin's separation theorem, with an induction on the dimension. As an application, we obtain Borel measurable selections of subgradients for parameter-dependent finite-dimensional convex functions, outside the scope of the standard normal integral framework. |
| title | Borel selection of dominating hyperplanes |
| topic | Logic Functional Analysis Optimization and Control |
| url | https://arxiv.org/abs/2603.19973 |