scholarly journals Nowhere differentiable intrinsic Lipschitz graphs

2021 ◽  
Author(s):  
Antoine Julia ◽  
Sebastiano Nicolussi Golo ◽  
Davide Vittone
Keyword(s):  
Lipschitz Graphs

scholarly journals Plenty of big projections imply big pieces of Lipschitz graphs

2021 ◽  
Author(s):  
Tuomas Orponen
Keyword(s):  
Lipschitz Graphs ◽  
Regular Sets ◽  
Big Pieces

AbstractI prove that closed n-regular sets $$E \subset {\mathbb {R}}^{d}$$ E ⊂ R d with plenty of big projections have big pieces of Lipschitz graphs. In particular, these sets are uniformly n-rectifiable. This answers a question of David and Semmes from 1993.

scholarly journals Two examples related to conical energies

2022 ◽  
Vol 47 (1) ◽  
pp. 261-281
Author(s):  
Damian Dąbrowski
Keyword(s):  
Singular Integral ◽  
Recent Article ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
The Other ◽  
Independent Interest ◽  
Sufficient Condition ◽  
Lipschitz Graphs ◽  
Big Pieces

In a recent article (2021) we introduced and studied conical energies. We used them to prove three results: a characterization of rectifiable measures, a characterization of sets with big pieces of Lipschitz graphs, and a sufficient condition for boundedness of nice singular integral operators. In this note we give two examples related to sharpness of these results. One of them is due to Joyce and Mörters (2000), the other is new and could be of independent interest as an example of a relatively ugly set containing big pieces of Lipschitz graphs.

scholarly journals AN INTEGRATION APPROACH TO THE TOEPLITZ SQUARE PEG PROBLEM

2017 ◽  
Vol 5 ◽  
Author(s):  
TERENCE TAO
Keyword(s):  
Simple Closed Curve ◽  
Closed Curve ◽  
Finite Sets ◽  
Real Numbers ◽  
Integration Approach ◽  
Sign Patterns ◽  
Lipschitz Constants ◽  
Lipschitz Graphs ◽  
Periodic Version ◽  
Rule Out

The ‘square peg problem’ or ‘inscribed square problem’ of Toeplitz asks if every simple closed curve in the plane inscribes a (nondegenerate) square, in the sense that all four vertices of that square lie on the curve. By a variety of arguments of a ‘homological’ nature, it is known that the answer to this question is positive if the curve is sufficiently regular. The regularity hypotheses are needed to rule out the possibility of arbitrarily small squares that are inscribed or almost inscribed on the curve; because of this, these arguments do not appear to be robust enough to handle arbitrarily rough curves. In this paper, we augment the homological approach by introducing certain integrals associated to the curve. This approach is able to give positive answers to the square peg problem in some new cases, for instance if the curve is the union of two Lipschitz graphs $f$, $g:[t_{0},t_{1}]\rightarrow \mathbb{R}$ that agree at the endpoints, and whose Lipschitz constants are strictly less than one. We also present some simpler variants of the square problem which seem particularly amenable to this integration approach, including a periodic version of the problem that is not subject to the problem of arbitrarily small squares (and remains open even for regular curves), as well as an almost purely combinatorial conjecture regarding the sign patterns of sums $y_{1}+y_{2}+y_{3}$ for $y_{1},y_{2},y_{3}$ ranging in finite sets of real numbers.

scholarly journals Connect the dots: how many random points can a regular curve pass through?

2005 ◽  
Vol 37 (03) ◽  
pp. 571-603 ◽  
Author(s):  
Ery Arias-Castro ◽  
David L. Donoho ◽  
Xiaoming Huo ◽  
Craig A. Tovey
Keyword(s):  
Image Processing ◽  
Point Clouds ◽  
Regular Curve ◽  
Smooth Functions ◽  
Higher Dimensional ◽  
Random Location ◽  
Pass Through ◽  
Lipschitz Graphs

Given a class Γ of curves in [0, 1]2, we ask: in a cloud of n uniform random points, how many points can lie on some curve γ ∈ Γ? Classes studied here include curves of length less than or equal to L, Lipschitz graphs, monotone graphs, twice-differentiable curves, and graphs of smooth functions with m-bounded derivatives. We find, for example, that there are twice-differentiable curves containing as many as O P (n 1/3) uniform random points, but not essentially more than this. More generally, we consider point clouds in higher-dimensional cubes [0, 1] d and regular hypersurfaces of specified codimension, finding, for example, that twice-differentiable k-dimensional hypersurfaces in R d may contain as many as O P (n k/(2d-k)) uniform random points. We also consider other notions of ‘incidence’, such as curves passing through given location/direction pairs, and find, for example, that twice-differentiable curves in R 2 may pass through at most O P (n 1/4) uniform random location/direction pairs. Idealized applications in image processing and perceptual psychophysics are described and several open mathematical questions are identified for the attention of the probability community.

scholarly journals Anisotropic mean curvature flow of Lipschitz graphs and convergence to self-similar solutions

2021 ◽  
Author(s):  
Annalisa Cesaroni ◽  
Heiko Kröner ◽  
Matteo Novaga
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Time Behavior ◽  
Long Time ◽  
Self Similar ◽  
The Stability ◽  
Lipschitz Graphs ◽  
Similar Solutions ◽  
Anisotropic Mean Curvature

We consider the anisotropic mean curvature flow of entire Lipschitz graphs. We prove existence and uniqueness of expanding self-similar solutions which are asymptotic to a prescribed cone, and we characterize the long time behavior of solutions, after suitable rescaling, when the initial datum is a sublinear perturbation of a cone. In the case of regular anisotropies, we prove the stability of self-similar solutions asymptotic to strictly mean convex cones, with respect to perturbations vanishing at infinity. We also show the stability of hyperplanes, with a proof which is novel also for the isotropic mean curvature flow.

scholarly journals On rectifiable measures in Carnot groups: representation

2021 ◽  
Vol 61 (1) ◽  
Author(s):  
Gioacchino Antonelli ◽  
Andrea Merlo
Keyword(s):  
Large Class ◽  
Hausdorff Measure ◽  
Carnot Group ◽  
Lipschitz Functions ◽  
Carnot Groups ◽  
Geodesic Sphere ◽  
Area Formula ◽  
Definition Of ◽  
Lipschitz Graphs ◽  
Almost All

AbstractThis paper deals with the theory of rectifiability in arbitrary Carnot groups, and in particular with the study of the notion of $$\mathscr {P}$$ P -rectifiable measure. First, we show that in arbitrary Carnot groups the natural infinitesimal definition of rectifiabile measure, i.e., the definition given in terms of the existence of flat tangent measures, is equivalent to the global definition given in terms of coverings with intrinsically differentiable graphs, i.e., graphs with flat Hausdorff tangents. In general we do not have the latter equivalence if we ask the covering to be made of intrinsically Lipschitz graphs. Second, we show a geometric area formula for the centered Hausdorff measure restricted to intrinsically differentiable graphs in arbitrary Carnot groups. The latter formula extends and strengthens other area formulae obtained in the literature in the context of Carnot groups. As an application, our analysis allows us to prove the intrinsic $$C^1$$ C 1 -rectifiability of almost all the preimages of a large class of Lipschitz functions between Carnot groups. In particular, from the latter result, we obtain that any geodesic sphere in a Carnot group equipped with an arbitrary left-invariant homogeneous distance is intrinsic $$C^1$$ C 1 -rectifiable.

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