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.
Categories: math.CA math.FA