Stabilization conjecture for iterated balanced refinements

Let $({\mathcal M},\rho)$ be a pseudometric space, let $X$ be a Banach space, and let $m$ be a fixed positive integer. Set $\ell=\ell(m,X)=\min\{m+1,\dim X\}$ . For a set-valued mapping $F:{\mathcal M}\to{\mathcal K}_m(X)$ , define

F^{[0]}=F,\qquad F^{[k+1]}(x)=\bigcap_{z\in{\mathcal M}}\left[F^{[k]}(z)+\lambda_{k+1}\rho(x,z)B_X\right].

Stabilization conjecture. There exist $\ell+1$ numbers $\lambda_1,\ldots,\lambda_{\ell+1}\ge1$ such that, for every $F$ satisfying the finiteness-principle hypothesis that each restriction to a subset of cardinality at most $N(m,X)$ has a Lipschitz selection with seminorm at most $1$ , one has

F^{[\ell]}(x)\ne\emptyset\qquad\text{and}\qquad F^{[\ell+1]}(x)=F^{[\ell]}(x)

for all $x\in{\mathcal M}$ . This asserts that the balanced-refinement process becomes nonempty and stabilizes after finitely many steps under the local selection hypothesis. The formulation is equivalent to requiring the terminal refinement to be a Lipschitz core, but expresses the claim without using the notion of a core.

The stabilization conjecture for iterated balanced refinements

Let $({\mathcal M},\rho)$ be a pseudometric space, let $X$ be a Banach space, and let $m$ be a fixed positive integer. Set

\ell=\ell(m,X)=\min\{m+1,\dim X\}.

For a sequence $\vec{\lambda}=\{\lambda_k:1\le k\le\ell+1\}$ , construct the iterated balanced refinements $F^{[k]}$ by

F^{[k+1]}(x)=\bigcap_{y\in{\mathcal M}}\left[F^{[k]}(y)+\lambda_{k+1}\rho(x,y)B_X\right].

The stabilization conjecture. There exists a sequence $\vec{\lambda}=\{\lambda_k:1\le k\le\ell+1\}$ with every $\lambda_k\ge1$ such that, for every set-valued mapping $F:{\mathcal M}\to{\mathcal K}_m(X)$ satisfying the hypothesis of the Finiteness Principle, one has

F^{[\ell]}(x)\ne\emptyset\quad\text{and}\quad F^{[\ell+1]}(x)=F^{[\ell]}(x)

for all $x\in{\mathcal M}$ . The claim predicts that the balanced-refinement process both remains nonempty through order $\ell$ and stabilizes at that order for every mapping satisfying the local finiteness hypothesis. The source gives no evidence of resolution.

source: Pavel Shvartsman, “The Core of a 2-Dimensional Set-Valued Mapping. Existence Criteria and Efficient Algorithms for Lipschitz Selections of Low Dimensional Set-Valued Mappings”, arXiv:2010.04540 (2021).

Stabilization conjecture for iterated balanced refinements

The stabilization conjecture for iterated balanced refinements

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