Displaying similar documents to “Directional properties of sets definable in o-minimal structures”

On lower Lipschitz continuity of minimal points

Ewa M. Bednarczuk (2000)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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In this paper we investigate the lower Lipschitz continuity of minimal points of an arbitrary set A depending upon a parameter u . Our results are formulated with the help of the modulus of minimality. The crucial requirement which allows us to derive sufficient conditions for lower Lipschitz continuity of minimal points is that the modulus of minimality is at least linear. The obtained results can be directly applied to stability analysis of vector optimization problems.

Bi-Lipschitz trivialization of the distance function to a stratum of a stratification

Adam Parusiński (2005)

Annales Polonici Mathematici

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Given a Lipschitz stratification 𝒳 that additionally satisfies condition (δ) of Bekka-Trotman (for instance any Lipschitz stratification of a subanalytic set), we show that for every stratum N of 𝒳 the distance function to N is locally bi-Lipschitz trivial along N. The trivialization is obtained by integration of a Lipschitz vector field.

Bi-Lipschitz Bijections of Z

Itai Benjamini, Alexander Shamov (2015)

Analysis and Geometry in Metric Spaces

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It is shown that every bi-Lipschitz bijection from Z to itself is at a bounded L1 distance from either the identity or the reflection.We then comment on the group-theoretic properties of the action of bi-Lipschitz bijections.

Michael's theorem for Lipschitz cells in o-minimal structures

Małgorzata Czapla, Wiesław Pawłucki (2016)

Annales Polonici Mathematici

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A version of Michael's theorem for multivalued mappings definable in o-minimal structures with M-Lipschitz cell values (M a common constant) is proven. Uniform equi-LCⁿ property for such families of cells is checked. An example is given showing that the assumption about the common Lipschitz constant cannot be omitted.

A decomposition of a set definable in an o-minimal structure into perfectly situated sets

Wiesław Pawłucki (2002)

Annales Polonici Mathematici

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A definable subset of a Euclidean space X is called perfectly situated if it can be represented in some linear system of coordinates as a finite union of (graphs of) definable 𝓒¹-maps with bounded derivatives. Two subsets of X are called simply separated if they satisfy the Łojasiewicz inequality with exponent 1. We show that every closed definable subset of X of dimension k can be decomposed into a finite family of closed definable subsets each of which is perfectly situated and such...

Local Lipschitz continuity of solutions of non-linear elliptic differential-functional equations

Pierre Bousquet (2007)

ESAIM: Control, Optimisation and Calculus of Variations

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The object of this paper is to prove existence and regularity results for non-linear elliptic differential-functional equations of the form div a ( ∇ u ) + F [ u ] ( x ) = 0 , over the functions u ∈ W 1 , 1 ( Ω ) that assume given boundary values on ∂Ω. The vector field a : ℝ n → ℝ n satisfies an ellipticity condition and for a fixed denotes a non-linear functional of In considering the same problem, Hartman and Stampacchia [ (1966) 271–310] have obtained existence results in the space of uniformly Lipschitz continuous functions...

A note on Lipschitz isomorphisms in Hilbert spaces

Dean Ives (2010)

Commentationes Mathematicae Universitatis Carolinae

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We show that the following well-known open problems on existence of Lipschitz isomorphisms between subsets of Hilbert spaces are equivalent: Are balls isomorphic to spheres? Is the whole space isomorphic to the half space?

Hölder regularity of two-dimensional almost-minimal sets in ℝ n

Guy David (2009)

Annales de la faculté des sciences de Toulouse Mathématiques

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We give a different and probably more elementary proof of a good part of Jean Taylor’s regularity theorem for Almgren almost-minimal sets of dimension 2 in ℝ 3 . We use this opportunity to settle some details about almost-minimal sets, extend a part of Taylor’s result to almost-minimal sets of dimension 2 in ℝ n , and give the expected characterization of the closed sets E of dimension 2 in ℝ 3 that are minimal, in the sense that H 2 ( E ∖ F ) ≤ H 2 ( F ∖ E ) for every closed set F such that there is a bounded set B so...

Geometry of Lipschitz percolation

G. R. Grimmett, A. E. Holroyd (2012)

Annales de l'I.H.P. Probabilités et statistiques

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We prove several facts concerning Lipschitz percolation, including the following. The critical probability L for the existence of an open Lipschitz surface in site percolation on ℤ with ≥ 2 satisfies the improved bound L ≤ 1 − 1/[8( − 1)]. Whenever > L, the height of the lowest Lipschitz surface above the origin has an exponentially decaying tail. For sufficiently close to 1, the connected regions of ℤ−1 above which the surface has height 2 or more exhibit stretched-exponential...