On Metric Regularity and the Boundary of the Feasible Set in Linear Optimization

This paper deals with semi-infinite linear inequality systems in ? ⁿ and studies the stability of the boundary of their feasible sets. We analyze the equivalence between the metric regularity of the inverse of the boundary set mapping, $\mathcal{N}$ , and the stability of the feasible set mapping in the sense of the maintenance of the consistency. In doing this we provide operational formulae for distances from points to some useful sets. We also include relationships between the regularity moduli corresponding to the mappings $\mathcal{N}$ and the inverse, $\mathcal{M}$ , of the feasible set mapping, and prove their equality for finite systems and some special cases in the semi-infinite framework. Moreover, we provide conditions to assure that the metric regularity of $\mathcal{N}$ is equivalent to the lower semi-continuity of the boundary set mapping, which is important because the latter property has many characterizations. Since the boundary of a feasible set may not be convex, we cannot make use of the general theory for mappings with convex graph, as for example, the Robinson–Ursescu theorem.     

Construction of {\mathcal L}^{{p}}-strong Feller Processes via Dirichlet Forms and Applications to Elliptic Diffusions

We provide a general construction scheme for $\mathcal L^p$ -strong Feller processes on locally compact separable metric spaces. Starting from a regular Dirichlet form and specified regularity assumptions, we construct an associated semigroup and resolvent of kernels having the $\mathcal L^p$ -strong Feller property. They allow us to construct a process which solves the corresponding martingale problem for all starting points from a known set, namely the set where the regularity assumptions hold. We apply this result to construct elliptic diffusions having locally Lipschitz matrix coefficients and singular drifts on general open sets with absorption at the boundary. In this application elliptic regularity results imply the desired regularity assumptions.     

Existence and regularity results for the Steiner problem

Given a complete metric space X and a compact set ${C\subset X}$ , the famous Steiner (or minimal connection) problem is that of finding a set S of minimum length (one-dimensional Hausdorff measure ${\mathcal H^1)}$ ) among the class of sets $$\mathcal{S}t(C) \,:=\{S\subset X\colon S \cup C \,{\rm is connected}\}.$$ In this paper we provide conditions on existence of minimizers and study topological regularity results for solutions of this problem. We also study the relationships between several similar variants of the Steiner problem. At last, we provide some applications to locally minimal sets.     

Singular integral operators with operator-valued kernels,and extrapolation of maximal regularity into rearrangement invariant Banach function spaces

We prove two extrapolation results for singular integral operators with operator-valued kernels, and we apply these results in order to obtain the following extrapolation of L ^p -maximal regularity: if an autonomous Cauchy problem on a Banach space has L ^p -maximal regularity for some \({p \in (1,\infty )}\) , then it has \({\mathbb{E}_w}\) -maximal regularity for every rearrangement invariant Banach function space \({\mathbb{E}}\) with Boyd indices \({1 < p_\mathbb{E} \leq q_\mathbb{E} < \infty}\) and every Muckenhoupt weight \({w \in A_{p \mathbb{E}}}\) . We prove a similar result for nonautonomous Cauchy problems on the line.     

On the Completions of the Spaces of Metrics on an Open Manifold II

Given a set X we construct a metric ρ on the set $ (\cal S)(X) $ of semi-metrics on X. We prove that ρ is complete and that a variety of interesting subsets of $ (\cal S)(X) $ are closed, giving rise to complete metric spaces of semi-metrics. In the second part we generalize this to a result about finite separating families of semi-metrics. In the third part of the paper we apply the results from the first part by constructing canonical metrics on spaces of riemannian metrics on an open manifold, which metricize some of the uniform structures defined in [3]. Finally we give some directions for possible applications.     

Decomposition of Geodesics in the Wasserstein Space and the Globalization Problem

We will prove a decomposition for Wasserstein geodesics in the following sense: let (X, d, m) be a non-branching metric measure space verifying ${\mathsf{CD}_{loc}(K,N)}$ or equivalently ${\mathsf{CD}^{*}(K,N)}$ . We prove that every geodesic ${\mu_{t}}$ in the L ²-Wasserstein space, with ${\mu_{t} \ll m}$ , is decomposable as the product of two densities, one corresponding to a geodesic with support of codimension one verifying ${\mathsf{CD}^{*}(K,N-1)}$ , and the other associated with a precise one dimensional measure, provided the length map enjoys local Lipschitz regularity. The motivation for our decomposition is in the use of the component evolving like ${\mathsf{CD}^{*}}$ in the globalization problem. For a particular class of optimal transportation we prove the linearity in time of the other component, obtaining therefore the global ${\mathsf{CD}(K,N)}$ for ${\mu_{t}}$ . The result can be therefore interpret as a globalization theorem for ${\mathsf{CD}(K,N)}$ for this class of optimal transportation, or as a “self-improving property” for ${\mathsf{CD}^{*}(K,N)}$ . Assuming more regularity, namely in the setting of infinitesimally strictly convex metric measure space, the one dimensional density is the product of two differentials giving more insight on the density decomposition.     

Nonlinear regularity models

The paper studies regularity properties of set-valued mappings between metric spaces. In the context of metric regularity, nonlinear models correspond to nonlinear dependencies of estimates of error bounds in terms of residuals. Among the questions addressed in the paper are equivalence of the corresponding concepts of openness and “pseudo-Hölder” behavior, general and local regularity criteria with special emphasis on “regularity of order $k$ ”, for local settings, and variational methods to extimate regularity moduli in case of length range spaces. The majority of the results presented in the paper are new.     

Bounds for graph regularity and removal lemmas

We show, for any positive integer k, that there exists a graph in which any equitable partition of its vertices into k parts has at least ck ²/log^* k pairs of parts which are not ${\epsilon}$ -regular, where ${c,\epsilon >0 }$ are absolute constants. This bound is tight up to the constant c and addresses a question of Gowers on the number of irregular pairs in Szemerédi’s regularity lemma. In order to gain some control over irregular pairs, another regularity lemma, known as the strong regularity lemma, was developed by Alon, Fischer, Krivelevich, and Szegedy. For this lemma, we prove a lower bound of wowzer-type, which is one level higher in the Ackermann hierarchy than the tower function, on the number of parts in the strong regularity lemma, essentially matching the upper bound. On the other hand, for the induced graph removal lemma, the standard application of the strong regularity lemma, we find a different proof which yields a tower-type bound. We also discuss bounds on several related regularity lemmas, including the weak regularity lemma of Frieze and Kannan and the recently established regular approximation theorem. In particular, we show that a weak partition with approximation parameter ${\epsilon}$ may require as many as ${2^{\Omega}(\epsilon^{-2})}$ parts. This is tight up to the implied constant and solves a problem studied by Lovász and Szegedy.     

Tree metrics and edge-disjoint S-paths

Given an undirected graph \(G=(V,E)\) with a terminal set \(S \subseteq V\) , a weight function on terminal pairs, and an edge-cost \(a: E \rightarrow \mathbf{Z}_+\) , the \(\mu \) -weighted minimum-cost edge-disjoint \(S\) -paths problem ( \(\mu \) -CEDP) is to maximize \(\sum \nolimits _{P \in \mathcal{P}} \mu (s_P,t_P) - a(P)\) over all edge-disjoint sets \(\mathcal{P}\) of \(S\) -paths, where \(s_P,t_P\) denote the ends of \(P\) and \(a(P)\) is the sum of edge-cost \(a(e)\) over edges \(e\) in \(P\) . Our main result is a complete characterization of terminal weights \(\mu \) for which \(\mu \) -CEDP is tractable and admits a combinatorial min–max theorem. We prove that if \(\mu \) is a tree metric, then \(\mu \) -CEDP is solvable in polynomial time and has a combinatorial min–max formula, which extends Mader’s edge-disjoint \(S\) -paths theorem and its minimum-cost generalization by Karzanov. Our min–max theorem includes the dual half-integrality, which was earlier conjectured by Karzanov for a special case. We also prove that \(\mu \) -EDP, which is \(\mu \) -CEDP with \(a = 0\) , is NP-hard if \(\mu \) is not a truncated tree metric, where a truncated tree metric is a weight function represented as pairwise distances between balls in a tree. On the other hand, \(\mu \) -CEDP for a truncated tree metric \(\mu \) reduces to \(\mu '\) -CEDP for a tree metric \(\mu '\) . Thus our result is best possible unless P = NP. As an application, we obtain a good approximation algorithm for \(\mu \) -EDP with “near” tree metric \(\mu \) by utilizing results from the theory of low-distortion embedding.     

On Convergent Interpolatory Subdivision Schemes in Riemannian Geometry

We show the convergence (for all input data) of refinement rules in Riemannian manifolds which are analogous to the linear four-point scheme and similar univariate interpolatory schemes, and which are generalized to the Riemannian setting by the so-called log/exp analogy. For this purpose, we use a lemma on the Hölder regularity of limits of contractive refinement schemes in metric spaces. In combination with earlier results on smoothness of limits, we settle the question of existence of interpolatory refinement rules intrinsic to Riemannian geometry which have \(C^r\) limits for all input data, for \(r \le 3\) . We further establish well-definedness of the reconstruction procedure of “interpolatory” multiscale transforms intrinsic to Riemannian geometry.     

Inductive topological Hausdorff dimensions and fibers of generic continuous functions

In an earlier paper Buczolich, Elekes and the author introduced a new concept of dimension for metric spaces, the so called topological Hausdorff dimension. They proved that it is precisely the right notion to describe the Hausdorff dimension of the level sets of the generic real-valued continuous function (in the sense of Baire category) defined on a compact metric space $K$ . The goal of this paper is to determine the Hausdorff dimension of the fibers of the generic continuous function from $K$ to $\mathbb {R}^n$ . In order to do so, we define the $n$ th inductive topological Hausdorff dimension, $\dim _{t^nH} K$ . Let $\dim _H K,\,\dim _t K$ and $C_n(K)$ denote the Hausdorff and topological dimension of $K$ and the Banach space of the continuous functions from $K$ to $\mathbb {R}^n$ . We show that $\sup _{y\in \mathbb {R}^n} \dim _{H}f^{-1}(y) = \dim _{t^nH} K -n$ for the generic $f \in C_n(K)$ , provided that $\dim _t K\ge n$ , otherwise every fiber is finite. In order to prove the above theorem we give some equivalent definitions for the inductive topological Hausdorff dimensions, which can be interesting in their own right. Here we use techniques coming from the theory of topological dimension. We show that the supremum is actually attained on the left hand side of the above equation. We characterize those compact metric spaces $K$ for which $\dim _{H} f^{-1}(y)=\dim _{t^nH}K-n$ for the generic $f\in C_n(K)$ and the generic $y\in f(K)$ . We also generalize a result of Kirchheim by showing that if $K$ is self-similar and $\dim _t K\ge n$ then $\dim _{H} f^{-1}(y)=\dim _{t^nH}K-n$ for the generic $f\in C_n(K)$ for every $y\in {{\mathrm{int}}}f(K)$ .     

Regularity of the Monge–Ampère equation in Besov’s spaces

Let $\mu = e^{-V} \ dx$ be a probability measure and $T = \nabla \Phi $ be the optimal transportation mapping pushing forward $\mu $ onto a log-concave compactly supported measure $\nu = e^{-W} \ dx$ . In this paper, we introduce a new approach to the regularity problem for the corresponding Monge–Ampère equation $e^{-V} = \det D^2 \Phi \cdot e^{-W(\nabla \Phi )}$ in the Besov spaces $W^{\gamma ,1}_{loc}$ . We prove that $D^2 \Phi \in W^{\gamma ,1}_{loc}$ provided $e^{-V}$ belongs to a proper Besov class and $W$ is convex. In particular, $D^2 \Phi \in L^p_{loc}$ for some $p>1$ . Our proof does not rely on the previously known regularity results.     

Existence of solutions for a nonlinear system of parabolic equations with gradient flow structure

We prove global existence of weak solutions of a variant of the parabolic-parabolic Keller–Segel model for chemotaxis on the whole space \({{\mathbb {R}}^d}\) for \(d\ge 3\) with a supercritical porous-medium diffusion exponent and an external drift. The structure of the equations allow the chemotactic drift to be seen both as attraction and repulsion. The method of proof relies on the inherent gradient flow structure of this system with respect to a coupled Wasserstein- \(L^2\) metric. Additional regularity estimates are derived from the dissipation of an entropy functional.     

A sequential implicit function theorem for the chords iteration

In this paper we study the local convergence of the method $$0 \in f\left( {p,x_k } \right) + A\left( {x_{k + 1} - x_k } \right) + F\left( {x_{k + 1} } \right),$$ in order to find the solution of the generalized equation $$find x \in X such that 0 \in f\left( {p,x} \right) + F\left( x \right).$$ We first show that under the strong metric regularity of the linearization of the associated mapping and some additional assumptions regarding dependence on the parameter and the relation between the operator A and the Jacobian $\nabla _x f\left( {\bar p,\bar x} \right)$ , we prove linear convergence of the method which is uniform in the parameter p. Then we go a step further and obtain a sequential implicit function theorem describing the dependence of the set of sequences of iterates of the parameter.     

Positive definite metric spaces

Magnitude is a numerical invariant of finite metric spaces, recently introduced by Leinster, which is analogous in precise senses to the cardinality of finite sets or the Euler characteristic of topological spaces. It has been extended to infinite metric spaces in several a priori distinct ways. This paper develops the theory of a class of metric spaces, positive definite metric spaces, for which magnitude is more tractable than in general. Positive definiteness is a generalization of the classical property of negative type for a metric space, which is known to hold for many interesting classes of spaces. It is proved that all the proposed definitions of magnitude coincide for compact positive definite metric spaces and further results are proved about the behavior of magnitude as a function of such spaces. Finally, some facts about the magnitude of compact subsets of $\ell _p^n$ for $p \le 2$ are proved, generalizing results of Leinster for $p=1,2$ using properties of these spaces which are somewhat stronger than positive definiteness.     

On the stability of \varphi -uniform domains

We study two metrics, the quasihyperbolic metric and the distance ratio metric of a subdomain $G \subset {\mathbb R}^n$ . In the sequel, we investigate a class of domains, so called $\varphi $ -uniform domains, defined by the property that these two metrics are comparable with respect to a homeomorphism $\varphi $ from $[0,\infty )$ to itself. Finally, we discuss a number of stability properties of $\varphi $ -uniform domains. In particular, we show that the class of $\varphi $ -uniform domains is stable in the sense that removal of a geometric sequence of points from a $\varphi $ -uniform domain yields a $\varphi _1$ -uniform domain.     

The structures of Hausdorff metric in non-Archimedean spaces

For non-Archimedean spaces X and Y, let $\mathcal{M}_\flat \left( X \right)$ , $\mathfrak{M}\left( {V \to W} \right)$ and $\mathfrak{D}_\flat \left( {X,Y} \right)$ be the ballean of X (the family of the balls in X), the space of mappings from X to Y, and the space of mappings from the ballean of X to Y, respectively. By studying explicitly the Hausdorff metric structures related to these spaces, we construct several families of new metric structures (e.g., $\hat \rho _u$ , $\hat \beta _{X,Y}^\lambda$ , $\hat \beta _{X,Y}^{ * \lambda }$ ) on the corresponding spaces, and study their convergence, structural relation, law of variation in the variable λ, including some normed algebra structure. To some extent, the class $\hat \beta _{X,Y}^\lambda$ is a counterpart of the usual Levy-Prohorov metric in the probability measure spaces, but it behaves very differently, and is interesting in itself. Moreover, when X is compact and Y = K is a complete non-Archimedean field, we construct and study a Dudly type metric of the space of K-valued measures on X.     

Sufficient conditions for regularity of area-preserving deformations

We discuss some basic regularity properties of the area-preserving deformations ${u:\Omega \mapsto \mathbb{R}^2}$ that have minimal elastic energy ${\int \limits_\Omega|\nabla u|^2}$ among a suitable class of admissible vectorfields defined on a smooth, bounded domain ${\Omega\subset \mathbb{R}^2}$ . Although we restrict ourselves to the quadratic stored energy function and 2-space, most of our results extend to three dimensional setting with convex stored energy function.     

Normalized Ricci flows and conformally compact Einstein metrics

In this paper, we investigate the behavior of the normalized Ricci flow on asymptotically hyperbolic manifolds. We show that the normalized Ricci flow exists globally and converges to an Einstein metric when starting from a non-degenerate and sufficiently Ricci pinched metric. More importantly we use maximum principles to establish the regularity of conformal compactness along the normalized Ricci flow including that of the limit metric at time infinity. Therefore we are able to recover the existence results in Graham and Lee (Adv Math 87:186–255, 1991), Lee (Fredholm Operators and Einstein Metrics on Conformally Compact Manifolds, 2006), and Biquard (Surveys in Differential Geometry: Essays on Einstein Manifolds, 1999) of conformally compact Einstein metrics with conformal infinities which are perturbations of that of given non-degenerate conformally compact Einstein metrics.