On an extension of Mazur's theorem on Lipschitz functions

The paper is partially supported by the Polish Committee for Scientific Research under grant no. 22009 91 02.     

On hilbertian subsets of finite metric spaces

The following result is proved: For everyε>0 there is aC(ε)>0 such that every finite metric space (X, d) contains a subsetY such that |Y|≧C(ε)log|X| and (Y, d _Y) embeds (1 +ε)-isomorphically into the Hilbert spacel ₂. The authors are grateful to Haim Wolfson for some discussions related to the content of this paper.     

Lipschitz representations of subsets of the cube

We show that for any class of uniformly bounded functions with a reasonable combinatorial dimension, the vast majority of small subsets of the -dimensional combinatorial cube cannot be represented as a Lipschitz image of a subset of , unless the Lipschitz constant is very large. We apply this result to the case when consists of linear functionals of norm at most one on a Hilbert space.

     

Quasiminima of the Lipschitz extension problem

In this paper, we extend the notion of quasiminimum to the framework of supremum functionals by studying the model case

Lipschitz factorization through subsets of Hilbert space

The Euclidean distortion of a metric space, a measure of how well the metric space can be embedded into a Hilbert space, is currently an active interdisciplinary research topic. We study the corresponding notion for mappings instead of spaces, which is that of Lipschitz factorization through subsets of Hilbert space. The main theorems are two characterizations of when a mapping admits such a factorization, both of them inspired by results dealing with linear factorizations through Hilbert space. The first is a nonlinear version of a classical theorem of Kwapień in terms of “dominated” sequences of vectors, whereas the second is a duality result by means of a tensor-product approach.     

On one-sided stabilizers of subsets of finite groups

It was initiated by the second author to investigate in which groups the left and right stabilizers of subsets have equal order. First we prove that if the left and right stabilizers of sets of prime power size are equal order then the group is supersolvable. We also characterize those 2-groups which satisfy this property for p = 2. We show that if in a finite group, the left and right stabilizers of sets of prime power size have equal order, then the commutator subgroup is abelian. Finally we characterize hamiltonian groups with the help of one-sided stabilizers. Received: 18 April 2005; revised 11 May 2005     

On skew field coproducts with a finite extension

It is shown that the field coproduct of any skew fieldE with a binomial (commutative) field extensionF/k overk can be expressed as a cyclic extension of a skew fieldK (theE-socle), itself the field coproduct of [F:k] copies ofE overk. Qua vector space the coproduct may also be expressed as a tensor product ofE andK overk. To the memory of Shimshon Amitsur     

Absolutely minimal Lipschitz extension of tree-valued mappings

We prove that every Lipschitz function from a subset of a locally compact length space to a metric tree has a unique absolutely minimal Lipschitz extension (AMLE). We relate these extensions to a stochastic game called Politics??a generalization of a game called Tug of War that has been used in Peres et?al. (J Am Math Soc 22(1):167?C210, 2009) to study real-valued AMLEs.     

On the chromatic number of finite systems of subsets

Mathematical Notes -     

On Burenkov's extension operator preserving Sobolev–Morrey spaces on Lipschitz domains

We prove that Burenkov's extension operator preserves Sobolev spaces built on general Morrey spaces, including classical Morrey spaces. The analysis concerns bounded and unbounded open sets with Lipschitz boundaries in the n‐dimensional Euclidean space.     

Stability of the Lipschitz extension property under metric transforms

We show that the linear and nonlinear Lipschitz extension properties of a metric space are not changed when the original metric is replaced by a new metric obtained by composition with an arbitrary concave function. Submitted: January 2001, Revised: February 2001.     

Metric extension operators,vertex sparsifiers and Lipschitz extendability

We study vertex cut and flow sparsifiers that were recently introduced by Moitra (2009), and Leighton and Moitra (2010). We improve and generalize their results. We give a new polynomial-time algorithm for constructing O(log k/ log log k) cut and flow sparsifiers, matching the best known existential upper bound on the quality of a sparsifier, and improving the previous algorithmic upper bound of O(log²k/ log log k). We show that flow sparsifiers can be obtained from linear operators approximating minimum metric extensions. We introduce the notion of (linear) metric extension operators, prove that they exist, and give an exact polynomialtime algorithm for finding optimal operators.     

Zero asymptotic Lipschitz distance and finite Gromov-Hausdorff distance

We give an example which shows that the Burago's bounded distance theorem does not hold in a non-intrinsic metric case. The argument is based on the classical answer to the densest circle packing problem in R2.