Related papers: Jordan curves inscribe a positive measure of recta…

We show that if $\gamma$ is a Jordan curve in $\mathbb{R}^2$ which is close to a $C^2$ Jordan curve $\beta$ in $\mathbb{R}^2$, then $\gamma$ contains an inscribed square. In particular, if $\kappa > 0$ is the maximum unsigned curvature of…

Let $J$ be a simple closed curve in $\mathbb R^{k}$ $(k\geq2)$ that is differentiable with non-zero derivative at a point $A_0\in J$. For a tuple of positive reals $a_1,\cdots,a_n$ $(n\geq3)$, each of which is less than the sum of the…

We prove that for every smooth Jordan curve $\gamma$, if $X$ is the set of all $r \in [0,1]$ so that there is an inscribed rectangle in $\gamma$ of aspect ratio $\tan(r\cdot \pi/4)$, then the Lebesgue measure of $X$ is at least $1/3$. To do…

We construct a Jordan curve $\Gamma \subset \mathbb{C}$ so that for any rectifiable arc $\sigma$ with endpoints in distinct complementary components of $\Gamma$, $H^1(\sigma \cap \Gamma) > 0$.

In this paper we prove that if $\gamma$ is a Jordan curve on $\mathbb{S}^2$ then there is a smooth curve shortening flow defined on $(0,T)$ which converges to $\gamma$ in $\mathcal{C}^0$ as $t\to 0^+ $. Another perspective is that the…

We study the $\epsilon$-level sets of the signed distance function to a planar Jordan curve $\Gamma$, and ask what properties of $\Gamma$ ensure that the $\epsilon$-level sets are Jordan curves, or uniform quasicircles, or uniform chord-arc…

Let g be an arbitrary Jordan loop and let G denote the space of rectangles R which are inscribed in g in such a way that the cyclic order of the vertices of R is the same whether it is induced by R or by g. We prove that G contains a…

We consider Jordan curves of the form $\gamma=\cup_{j=1}^n \gamma_j$ on the Riemann sphere for which each $\gamma_j$ is a hyperbolic geodesic in $(\widehat{\mathbb C} \smallsetminus \gamma)\cup \gamma_j$. These Jordan curves are…

We use Batson's lower bound on the nonorientable slice genus of $(2n,2n-1)$-torus knots to prove that for any $n \geq 2$, every smooth Jordan curve has an inscribed rectangle of of aspect ratio $\tan(\frac{\pi k}{2n})$ for some $k\in…

The square-peg problem asks if every Jordan curve in the plane has four points which are the vertices of a square. The problem is open for continuous Jordan curves, but it has been resolved for various regularity classes of curves between…

If two Jordan curves in the plane have precisely one point in common, and there they do not properly cross, then the common point is called a {\em touching point}. The main result of this paper is a Crossing Lemma for simple curves: Let $X$…

The square peg problem asks whether every Jordan curve in the plane has four points which form a square. The problem has been resolved (positively) for various classes of curves, but remains open in full generality. We present two new…

Given a plane curve $\gamma: S^1\to \mathbb R^2$, we consider the problem of determining the minimal number $I(\gamma)$ of inflections which curves $\mbox{diff}(\gamma)$ may have, where $\mbox{diff}$ runs over the group of diffeomorphisms…

For every smooth Jordan curve $\gamma$ and rectangle $R$ in the Euclidean plane, we show that there exists a rectangle similar to $R$ whose vertices lie on $\gamma$. The proof relies on Shevchishin's theorem that the Klein bottle does not…

Meyerson's Theorem says that all but at most 2 points of any Jordan loop are vertices of inscribed equilateral triangles. We show that for any Jordan loop there are uncountable many other triangle shapes for which this same result is true.…

Let $f$ be a meromorphic function with simply connected domain $G\subset\mathbb{C}$, and let $\Gamma\subset\mathbb{C}$ be a smooth Jordan curve. We call a component of $f^{-1}(\Gamma)$ in $G$ a $\Gamma$-$pseudo$-$lemniscate$ of $f$. In this…

We prove the three embeddedness results as follows. $({\rm i})$ Let $\Gamma_{2m+1}$ be a piecewise geodesic Jordan curve with $2m+1$ vertices in $\mathbb{R}^n$, where $m$ is an integer $\geq2$. Then the total curvature of…

For a Jordan domain in the plane the length metric space of points connected to an interior point by a curve of finite length is a CAT(0)space and Gromov hyperbolic. With respect to the cone topology, that space plus its boundary at…

We prove that for every smooth Jordan curve $\gamma \subset \mathbb{C}$ and for every set $Q \subset \mathbb{C}$ of six concyclic points, there exists a non-constant quadratic polynomial $p \in \mathbb{C}[z]$ such that $p(Q) \subset…

Our main result states that, under an exponential map whose Julia set is the whole complex plane, on each piecewise smooth Jordan curve there is a point whose orbit is dense. This has consequences for the boundaries of nice sets, used in…