Related papers: A dichotomy of sets via typical differentiability

The set of answers to a query may be very large, potentially overwhelming users when presented with the entire set. In such cases, presenting only a small subset of the answers to the user may be preferable. A natural requirement for this…

The main motivation of this paper arises from the study of Carnot-Carath\'eodory spaces, where the class of 1-rectifiable sets does not contain smooth non-horizontal curves; therefore a new definition of rectifiable sets including…

Let (X, d) be a quasi-convex, complete and separable metric space with reference probability measure m. We prove that the set of of real valued Lipschitz function with non zero point-wise Lipschitz constant m-almost everywhere is residual,…

The present paper develops two concepts of pointwise differentiability of higher order for arbitrary subsets of Euclidean space defined by comparing their distance functions to those of smooth submanifolds. Results include that…

We improve by an exponential factor the best known asymptotic upper bound for the density of sets avoiding 1 in Euclidean space. This result is obtained by a combination of an analytic bound that is an analogue of Lovasz theta number and of…

We apply the topology of convergence on compact sets to define unpredictable functions [5, 6]. The topology is metrizable and easy for applications with integral operators. To demonstrate the effectiveness of the approach, the existence and…

In this work we show that an $n$-dimensional Borel set in Euclidean $N$-space with finite integral Menger curvature is $n$-rectifiable, meaning that it can be covered by countably many images of Lipschitz continuous functions up to a null…

We construct a complete metric space $M$ of cardinality continuum such that every non-singleton closed separable subset of $M$ fails to be a Lipschitz retract of $M$. This provides a metric analogue to the various classical and recent…

We show that any non-zero Banach space with a separable dual contains a totally disconnected, closed and bounded subset S of Hausdorff dimension 1 such that every Lipschitz function on the space is Fr\'echet differentiable somewhere in S.

We prove that each discrete set in the Euclidean space that has bounded changes under every translation is a bounded perturbation of a square lattice, i.e., a uniformly spread set in the sense of Laszkovich. In particular, the support of…

The purpose of this note is to point out a simple consequence of some earlier work of the authors, "Hard Sard: Quantitative implicit function and extension theorems for Lipschitz maps". For $f$, a Lipschitz function from a Euclidean space…

We denote the local ``little" Lipschitz constant of a function $f: {{\mathbb R}}\to { {\mathbb R}}$ by $ {\mathrm{lip}}f$. In this paper we settle the following question: For which sets $E {\subset} { {\mathbb R}}$ is it possible to find a…

We construct a H\"older continuous function on the unit interval which coincides in uncountably (in fact continuum) many points with every function of total variation smaller than 1 passing through the origin. We say that a function with…

We present a general method of constructing an uncountable family of regular Borel measures on certain path spaces of Lipschitz functions having fixed Lipschitz constants. We use this method to give a definition of Lebesgue measure and…

We conjecture that whenever $M$ is a metric space of density at most continuum, then the space of Lipschitz functions is $w^*$-separable. We prove the conjecture for several classes of metric spaces including all the Banach spaces with a…

The classical Besicovitch projection theorem states that if a planar set $E$ with finite length is purely unrectifiable, then almost all orthogonal projections of $E$ have zero length. We prove a quantitative version of this result: if…

Entropy rate is a real valued functional on the space of discrete random sources which lacks a closed formula even for subclasses of sources which have intuitive parameterizations. A good way to overcome this problem is to examine its…

It was established in [8] that Lipschitz inf-compact functions are uniquely determined by their local slope and critical values. Compactness played a paramount role in this result, ensuring in particular the existence of critical points. We…

On a non-compact, smooth, connected, boundaryless, complete Riemannian manifold $(M,g)$, one can define its ideal boundary by rays (or equivalently, Busemann functions). From the viewpoint of Mather theory, boundary elements could be…

It is shown that a set in product of $n$ metrizable spaces is the discontinuity points set of some separately continuous function if and only if this set can be represented as the union of a sequence of $F_{\sigma}$-sets which are locally…