Minimizers of the Maximum Distance Problem via an Analyst's Traveling Salesperson Algorithm
We provide an upper and lower bound for the length of Maximum Distance Problem minimizers in terms of a finite scale geometric square sum.
math.MG↗
arXiv subjects
Silvia Ghinassi
Publications and source records attributed to Silvia Ghinassi.
A step towards the tensorization of Sobolev spaces
We prove that Sobolev spaces on Cartesian and warped products of metric spaces tensorize, only requiring that one of the factors is a doubling space supporting a Poincaré inequality.
math.MG↗
Self-similar sets and Lipschitz graphs
We investigate and quantify the distinction between rectifiable and purely unrectifiable 1-sets in the plane. That is, given that purely unrectifiable 1-sets always have null intersections with Lipschitz images, we ask whether these sets intersect with Lipschitz images at a dimension that is close to one. In an answer to this question, we show that one-dimensional attractors of iterated function systems that satisfy the open set condition have subsets of dimension arbitrarily close to one that can be covered by Lipschitz graphs. Moreover, the Lipschitz constant of such graphs depends explicitly on the difference between the dimension of the original set and the subset that intersects with the graph.
math.CA↗
Endpoint regularity for $2d$ Mumford-Shah minimizers: On a theorem of Andersson and Mikayelyan
We give an alternative proof of the regularity, up to the loose end, of minimizers, resp. critical points of the Mumford-Shah functional when they are sufficiently close to the cracktip, resp. they consist of a single arc terminating at an interior point.
math.AP↗
Sufficient conditions for $C^{1,α}$ parametrization and rectifiability
We say a measure is $C^{1,α}$ $d$-rectifiable if there is a countable union of $C^{1,α}$ $d$-surfaces whose complement has measure zero. We provide sufficient conditions for a Radon measure in $\mathbb{R}^n$ to be $C^{1,α}$ $d$-rectifiable, with $α\in (0,1]$. The conditions involve a Bishop-Jones type square function and all statements are quantitative in that the $C^{1,α}$ constants depend on such a function. Along the way we also give sufficient conditions for $C^{1,α}$ parametrizations for Reifenberg flat sets in terms of the same square function. Key tools for the proof come from David and Toro's Reifenberg parametrizations of sets with holes in the Hölder and Lipschitz categories.
math.MG↗
Menger curvatures and $C^{1,α}$ rectifiability of measures
We further develop the relationship between $β$-numbers and discrete curvatures to provide a new proof that under weak density assumptions, finiteness of the pointwise discrete curvature $\operatorname{curv}^α_{μ;2}(x,r)$ at $μ$- a.e. $x \in \mathbb{R}^{m}$ implies that $μ$ is $C^{1,α}$ $n$-rectifiable.
math.MG↗