arXiv subjects

Jessica Merhej

Publications and source records attributed to Jessica Merhej.

2 recordsLinked to original sources

arXiv2016

Poincar\'e-type Inequalities and Finding Good Parameterizations

A very important question in geometric measure theory is how geometric features of a set translate into analytic information about it. In 1960, E. R. Reifenberg proved that if a set is well approximated by planes at every point and at every scale, then the set is a bi-H\"older image of a plane. It is known today that Carleson-type conditions on these approximating planes guarantee a bi-Lipschitz parameterization of the set. In this paper, we consider an $n$-Ahlfors regular rectifiable set $M \subset \mathbb{R}^{n+d}$ that satisfies a Poincar\'{e}-type inequality involving the tangential derivative. Then, we show that a Carleson-type condition on the oscillations of the tangent planes of $M$ guarantees that $M$ is contained in a bi-Lipschitz image of an $n$-plane. We also explore the Poincar\'e-type inequality considered here and show that it is in fact equivalent to other Poincar\'e-type inequalities considered on general metric measure spaces.

math.MG↗

arXiv2015

On the Geometry of Rectifiable Sets with Carleson and Poincar\'e-type Conditions

A central question in geometric measure theory is whether geometric properties of a set translate into analytical ones. In 1960, E. R. Reifenberg proved that if an $n$-dimensional subset $M$ of $\mathbb{R}^{n+k}$ is well approximated by $n$-planes at every point and at every scale, then $M$ is a locally bi-H\"older image of an $n$-plane. Since then, Reifenberg's theorem has been refined in several ways in order to ensure that $M$ is a bi-Lipschitz image of an $n$-plane. In this paper, we show that a Carleson condition on the oscillation of the unit normal of an $n$-Ahlfors regular rectifiable subset $M$ of $\mathbb{R}^{n+1}$ satisfying a Poincar\'e-type inequality is sufficient to prove that $M$ is contained inside a bi-Lipschitz image of an $n$-dimensional affine subspace of $\mathbb{R}^{n+1}$. We also show that this Poincar\'e-type inequality encodes geometrical information about $M$, namely it implies that $M$ is quasiconvex.

math.CA↗