Lipschitz Selections Research Articles

On the regularity of selections and on Minty points of generalized monotone set-valued maps

In this paper we deal with set-valued maps T:X⇉X⁎ defined on a Banach space X, that are generalized monotone in the sense of Karamardian. Under various continuity assumptions on T, we investigate the regularity of suitable selections of the set-valued map R++T that shares with T the generalized monotonicity properties. In particular, we show that for every quasimonotone set-valued map T satisfying the Aubin property around (y,y⁎)∈gph(T) with y⁎≠0 there exist locally Lipschitz selections of R++T∖{0}. In the last part some notions of Minty points of T are introduced, and their relationship with zeros as well as effective zeros of T is discussed; a correlation is established between these concepts and the broader context of the generalized monotonicity of the map T.

On the core of a low dimensional set-valued mapping

Let \mathfrak{M}=(\mathcal{M},\rho) be a metric space and let X be a Banach space. Let F be a set-valued mapping from \mathcal{M} into the family \mathcal{K}_m(X) of all compact convex subsets of X of dimension at most m . The main result in our recent joint paper with Charles Fefferman (which is referred to as a “finiteness principle for Lipschitz selections”) provides efficient conditions for the existence of a Lipschitz selection of F , i.e., a Lipschitz mapping f\colon \mathcal{M}\to X such that f(x)\in F(x) for every x\in\mathcal{M} . We give new alternative proofs of this result in two special cases. When m=2 , we prove it for X=\mathbb{R}^{2} , and when m=1 we prove it for all choices of X . Both of these proofs make use of a simple reiteration formula for the “core” of a set-valued mapping F , i.e., for a mapping G\colon \mathcal{M}\to\mathcal{K}_m(X) which is Lipschitz with respect to the Hausdorff distance, and such that G(x)\subset F(x) for all x\in\mathcal{M} .

Existence of Lipschitz selections of the Steiner map

This paper is concerned with the problem of the existence of Lipschitz selections of the Steiner map , which associates with points of a Banach space the set of their Steiner points. The answer to this problem depends on the geometric properties of the unit sphere of , its dimension, and the number . For general conditions are obtained on the space under which admits no Lipschitz selection. When is finite dimensional it is shown that, if is even, the map has a Lipschitz selection if and only if is a finite polytope; this is not true if is odd. For the (single-valued) map is shown to be Lipschitz continuous in any smooth strictly-convex two-dimensional space; this ceases to be true in three-dimensional spaces. Bibliography: 21 titles.

On Stability of Fixed Points for Multi-Valued Mappings with an Application

This paper studies the stability of fixed points for multi-valued mappings in relation to selections. For multi-valued mappings admitting Michael selections, some examples are given to show that the fixed point mapping of these mappings are neither upper semi-continuous nor almost lower semi-continuous. Though the set of fixed points may be not compact for multi-valued mappings admitting Lipschitz selections, by finding sub-mappings of such mappings, the existence of minimal essential sets of fixed points is proved, and we show that there exists at least an essentially stable fixed point for almost all these mappings. As an application, we deduce an essentially stable result for differential inclusion problems.

Lipschitz selections of the diametric completion mapping in Minkowski spaces

We develop a constructive completion method in general Minkowski spaces, which successfully extends a completion procedure due to Bückner in two- and three-dimensional Euclidean spaces. We prove that this generalized Bückner completion is locally Lipschitz continuous, thus solving the problem of finding a continuous selection of the diametric completion mapping in finite dimensional normed spaces. The paper also addresses the study of an elegant completion procedure due to Maehara in Euclidean spaces, the natural setting of which are the spaces with a generating unit ball. We prove that, in these spaces, the Maehara completion is also locally Lipschitz continuous, besides establishing other geometric properties of this completion. The paper contains also new estimates of the (local) Lipschitz constants for the wide spherical hull.

Locally Lipschitz selections in Banach lattices

Let X be a Banach space while ( Y , ⪯ ) is a Banach lattice. We consider the class of “upper separated” set-valued functions F : X → 2 Y and investigate the problem of the existence of convex and locally Lipschitz selections of F . We discuss some applications of the selection results obtained in the paper to the theory of deterministic and stochastic differential inclusions.

1-Type Lipschitz selections in generalized 2-normed spaces

We shall introduce 1-type Lipschitz multifunctions from ℝ into generalized 2-normed spaces, and give some results about their 1-type Lipschitz selections.

The Whitney extension problem and Lipschitz selections of set-valued mappings in jet-spaces

We study a variant of the Whitney extension problem (1934) for the space C k,ω (R n ). We identify C k,ω (R n ) with a space of Lipschitz mappings from n into the space P k x R of polynomial fields on n equipped with a certain metric. This identification allows us to reformulate the Whitney problem for C k,ω (R n ) as a Lipschitz selection problem for set-valued mappings into a certain family of subsets of P k x R. We prove a Helly-type criterion for the existence of Lipschitz selections for such set-valued mappings defined on finite sets. With the help of this criterion, we improve estimates for finiteness numbers in finiteness theorems for C k,ω (R n ) due to C. Fefferman.

Lipschitz selections and stability for quasiconvex programs

Stability of ε-optimal solutions for quasiconvex programs is studied.

Lipschitz selections of set-valued mappings and Helly’s theorem

We prove a Helly-type theorem for the family of all m-dimensional convex compact subsets of a Banach space X. The result is formulated in terms of Lipschitz selections of set-valued mappings from a metric space (M, ρ) into this family. Let M be finite and let F be such a mapping satisfying the following condition: for every subset M′ ⊂ M consisting of at most 2m+1 points, the restriction F¦M′ of F to M′ has a selection fM′ (i. e., fM′(x) ∈ F(x) for all x ∈ M′) satisfying the Lipschitz condition ‖ƒM′(x) − ƒM′(y)‖X ≤ ρ(x, y), x, y ∈ M′. Then F has a Lipschitz selection ƒ: M → X such that ‖ƒ(x) − ƒ(y)‖X ≤ γρ(x,y), x, y ∈ M where γ is a constant depending only on m and the cardinality of M. We prove that in general, the upper bound of the number of points in M′, 2m+1, is sharp. If dim X = 2, then the result is true for arbitrary (not necessarily finite) metric space. We apply this result to Whitney’s extension problem for spaces of smooth functions. In particular, we obtain a constructive necessary and sufficient condition for a function defined on a closed subset of R 2 to be the restriction of a function from the Sobolev space W ∞ 2 (R 2).A similar result is proved for the space of functions on R 2 satisfying the Zygmund condition.

On Lipschitz selections of affine-set valued mappings

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 \( ({\cal M},\rho) \) into this family.¶Let F be such a mapping satisfying the following condition: for every subset \( {\cal M'} \subset {\cal M} \) consisting of at most 2k+1 points, the restriction \( F|_{\cal M'} \) of F to \( {\cal M'} \) has a selection \( f_{\cal M'}\,({\rm i.e.}\,f_{\cal M'}(x) \in F(x)\,{\rm for\,all}\,x\,\in {\cal M'}) \) satisfying the Lipschitz condition \( \parallel f_{\cal M'}(x) - f_{\cal M'}(y)\parallel\,\le \rho(x,y ),\,x,y \in {\cal M'} \). Then F has a Lipschitz selection \( f : {\cal M} \to H \) such that \( \parallel f(x) - f(y) \parallel\,\le \gamma \rho (x,y ),\,x,y \in {\cal M} \) where \( \gamma = \gamma(k) \) is a constant depending only on k. (The upper bound of the number of points in \( {\cal M'} \), 2k+1, is sharp.)¶The proof is based on a geometrical construction which allows us to reduce the problem to an extension property of Lipschitz mappings defined on subsets of metric trees.

Lipschitz Kernel of a Closed Set-Valued Map

This paper deals with Lipschitz selections of set-valued maps with closed graphs. First, we characterize Lipschitzianity of a closed set-valued map in the differential games framework in terms of a discriminating property of its graph. This allows us to consider the λ-Lipschitz kernel of a given set-valued map as the largest λ-Lipschitz closed set-valued map contained in the initial one, to derive an algorithm to compute the collection of Lipschitz selections, and to extend the Pasch–Hausdorff envelope to set-valued maps.

Erratum: Hoffman's Error Bound, Local Controllability, and Sensitivity Analysis

The proofs of Lemma 5.2 and Theorem 5.1 in [SIAM J. Control Optim., 38 (2000), pp. 947--970] have a common error which concerns the existence of (locally) Lipschitz selections. Indeed, Lemma 5.2 is true in the case where the multivalued mapping G is convex-valued, and in our case this assumption is not satisfied. So some modifications are needed. The statement of Theorem 5.1 (which does not use Lemma 5.2) is as follows.

The trace of jet space 𝐽^{𝑘}Λ^{𝜔} to an arbitrary closed subset of ℝⁿ

The classical Whitney extension theorem describes the trace J k | X J^k|_X of the space of k k -jets generated by functions from C k ( R n ) C^k(\mathbb R^n) to an arbitrary closed subset X ⊂ R n X\subset \mathbb R^n . It establishes existence of a bounded linear extension operator as well. In this paper we investigate a similar problem for the space C k Λ C^k\Lambda of functions whose higher derivatives satisfy the Zygmund condition with majorant ω \omega . The main result states that the vector function f → = ( f α : X → R ) | α | ≤ k \vec f=(f_\alpha \colon X\to \mathbb R)_{|\alpha |\le k} belongs to the corresponding trace space if the trace f → | Y \vec f|_Y to every subset Y ⊂ X Y\subset X of cardinality 3 ⋅ 2 ℓ 3\cdot 2^\ell , where ℓ = ( n + k − 1 k + 1 ) \ell =(\begin {smallmatrix}n+k-1 k+1\end {smallmatrix}) , can be extended to a function f Y ∈ C k Λ f_Y\in C^k\Lambda and sup Y | f Y | C k Λ ω > ∞ \sup _Y|f_Y|_{C^k\Lambda ^\omega }>\infty . The number 3 ⋅ 2 l 3\cdot 2^l generally speaking cannot be reduced. The Whitney theorem can be reformulated in this way as well, but with a two-pointed subset Y ⊂ X Y\subset X . The approach is based on the theory of local polynomial approximations and a result on Lipschitz selections of multivalued mappings.

Extensions of Lipschitz selections and an application to differential inclusions

WE ESTABLISH in one section of this paper, the existence of a Lipschitz extension of a Lipschitz selection of a Lipschitz convex-compact set-valued map into R”. That is, given a Lipschitz selection on a subdomain, it can be extended to a Lipschitz selection on the entire domain of definition of the set-valued map. The extension result is applied in the closing section. We show that under the minimal natural conditions, a Lipschitz solution of an autonomous Lipschitz differential inclusion, is a solution of an autonomous Lipschitz ordinary differential equation within the inclusion.