On Lipschitz selections of affine-set valued mappings

Abstract

We prove a Helly-type theorem for the family of all k-dimensional affine subsets of a Hilbert space H. The result is formulated in terms of Lipschitz selections of set-valued mappings from a metric space (M, ρ) into this family.Let F be such a mapping satisfying the following condition: for every subset M ⊂ M consisting of at most 2 k+1

Whitney’s extension problem for multivariate 𝐶^{1,𝜔}-functions

“…This result was first proved in [Sh1]; see also [Sh2]. In fact, the main Theorem 1.3 requires a generalization of Theorem 1.4 related to selection of set-valued mappings defined on metric graphs.…”

Finiteness Principles for Smooth Selection

“…(Recall that P 1 , P 2 are polynomials of degree at most m − 1.) Thanks to (27), (28), (30), (33), and the (C w , δ max )-convexity of Γ , we may apply Lemma 1 to the polynomials P 1 ,…”

“…In fact, P. Shvartsman in [24] showed that one can take k # = 2 d+1 in Theorem 2 and that the constant k # = 2 d+1 is sharp, see [26]. P. Shvartsman also showed that Theorem 2 remains valid when R D is replaced by a Hilbert space (see [27]) or a Banach space (see [29]).…”

Interpolation of data by smooth nonnegative functions

“…This paper is part of a literature on extension, interpolation, and selection of functions, going back to H. Whitney's seminal work [33], and including fundamental contributions by G. Glaeser [19], Y, Brudnyi and P. Shvartsman [4,[6][7][8][9][23][24][25][26][27][28][29][30][31], J. Wells [32], E. Le Gruyer [21], and E. Bierstone, P. Milman, and W. Paw lucki [1][2][3], as well as our own papers [10][11][12][13][14][15][16][17].…”