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Sharp finiteness principles for Lipschitz selections. (English) Zbl 1412.46044

The authors prove the finiteness theorem for Lipschitz selection problems, conjectured by Y. Brudnyi and P. Shvartsman [Int. Math. Res. Not. 1994, No. 3, 129–139 (1994; Zbl 0845.57022)]. This has previously been established by C. Fefferman et al. [Geom. Funct. Anal. 26, No. 2, 422–477 (2016; Zbl 1353.58004)] and Shvartsman in a sequence of papers in some particular cases.
The result is stated in full generality. Let \(\mathcal K_m(Y)\) stand for the family of nonempty compact convex subsets \(K\subset Y\) contained in an affine subspace of dimension at most \(m\). Given \(m\geq 1\) and \(\lambda>0\), a Banach space \(Y\) and a pseudo-metric space \((\mathcal M, \rho)\), if \(F:\mathcal M\to \mathcal K_m(Y)\) satisfies that \(F|_{\mathcal M'}\) has a Lipschitz selection \(f_{\mathcal M'}\) with Lipschitz constant bounded by \(\lambda\) for any \(\mathcal M'\subset \mathcal M\) consisting of at most \(2^{\dim(Y)}\) (resp., \(2^{m+1}\)) points in the case \(\dim(Y)\leq m\) (resp., \(\dim(Y) \ge m+1\)), then \(F\) also has a Lipschitz selection \(f\) with Lipschitz constant bounded by \(\gamma \lambda\), where \(\gamma\) depends only on \(m\). The cases \(m=1,2\) were previously proved by the authors. Also, the case \(Y=\mathbb R^2\) had already been shown by the second named author (see [P. Shvartsman, J. Geom. Anal. 12, No. 2, 289–324 (2002; Zbl 1031.52004)]), but the case \(Y=\mathbb R^n\) for metric spaces was still unknown, and it arose in connection with Whitney’s extension problem. Observe that, when the result is applied for the trivial distance \(\rho=0\), one obtains Helly’s classical theorem, however, as mentioned in the paper, the proof of the general case makes extensive use of Helly’s theorem. The result gives sharp “finiteness constants” in both the finite- or infinite-dimensional situation.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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