Web1 jul. 2024 · Theorem 4.1 Let Y be a metric space such that every closed ball in Y is compact. Then Y is ( R d, ℓ 1) -Kirszbraun if and only if Y is geodesically complete and ( 2 d, 2) -Helly. First, The recognition problem Below we give a polynomial time algorithm to decide whether a given graph is Z d -Kirszbraun. Web1.2 Theorem (Kirszbraun, Valentine) If X;Y are Hilbert spaces, AˆX, and f: A!Y is -Lipschitz, then fhas a -Lipschitz extension f : X!Y. See [Kirs], [Val], or [Fed, 2.10.43]. A generalization to metric spaces with curvature bounds was given in [LanS]. The next result characterizes the extendability of partially de ned Lip-
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Webin [Val45] famously showed how the Helly theorem can be used to obtain the Kirszbraun theorem. The connection between these two theorems is the key motivation behind this paper. Given metric spaces Xand Y, we say that Y is X-Kirszbraun if all AˆX, every 1-Lipschitz maps f : A! Y has a 1-Lipschitz extension from Ato X. In this notation, the ... Web18 dec. 2024 · However, in the case we present here, we can use Kirszbraun's theorem from nonlinear functional analysis [24], constructing G as a piecewise linear function by … happy birthday quotes for men
Kirszbraun
Webtheorem, namely the Kirszbraun theorem, which allows us to “invert” the dimensionality reduction map and argue the preservation of the cost of the Wasserstein barycenter under a general L p objective. For more details, see Section 3. Dimensionality reduction independent of k. While the JL lemma is known to be tight [LN16, Webis theorem was generalized for Hilbert spaces X,Y in place of Rn and Rm by F. A. Valentine [çý] in ÔÀ¥€, and the result is oŸen referred to as the Kirszbraun– Valentine theorem. e proof is rather nonconstructive, in the sense that it requires the use of Zorn’s lemma or transfinite induction at least in the nonseparable case. WebKirszbraun theorem In mathematics, specifically real analysis and functional analysis, the Kirszbraun theorem states that if " U " is a subset of some Hilbert space " H " 1, and " H " 2 is another Hilbert space, and : " f " : " U " → " H " 2 is a Lipschitz - continuous map, then there is a Lipschitz - continuous map : " F ": " H " 1 → " H " 2 happy birthday quotes for new friend