Finiteness Principles for Smooth Selection
Open Access Publications from the University of California

## Finiteness Principles for Smooth Selection

• Author(s): Fefferman, Charles
• Israel, Arie
• Luli, Garving K.
• et al.

## Published Web Location

https://arxiv.org/pdf/1511.04804.pdf
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Abstract

In this paper we prove finiteness principles for $C^{m}\left( \mathbb{R}^{n}, \mathbb{R}^{D}\right)$-selection, and for $C^{m-1,1}\left( \mathbb{R}^{n}, \mathbb{R}^{D}\right)$-selection, in particular providing a proof for a conjecture of Brudyni-Shvartsman (1994) on Lipschitz selections for the case when the domain is $X = \mathbb{R}^n$. Our results raise the hope that one can start to understand constrained interpolation problems in which e.g. the interpolating function $F$ is required to be nonnegative everywhere.

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