Borel selection of dominating hyperplanes

TL;DR

This paper establishes the existence of Borel measurable selections of dominating hyperplanes for families of finite-dimensional functions, using semi-analytic regularity assumptions and a novel combination of Borel insertion and induction techniques.

Contribution

It introduces a new measurable selection theorem for dominating hyperplanes under semi-analytic conditions, extending beyond standard normal integral frameworks.

Findings

Existence of Borel measurable selectors for dominating hyperplanes.

Application to measurable selection of subgradients for convex functions.

Extension beyond classical uniformisation theorems.

Abstract

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.