arXiv research

A regular $n$-gon inscribing a knot is a sequence of $n$ points on a knot, such that the distances between adjacent points are all the same. It is shown that any smooth knot is inscribed by a regular $n$-gon for any $n$.

Smooth knots with odd Conway polynomial terms have inscribed trefoils.

Study on non-orientable surfaces and inscribed rectangles in 4-manifolds.

Generalizing Milnor's result that an FTC (finite total curvature) knot has an isotopic inscribed polygon, we show that any two nearby knotted FTC graphs are isotopic by a small isotopy. We also show how to obtain sharper constants when the starting curve is smooth. We apply our main theorem to prove a limiting result f…

We establish a fundamental connection between smooth and polygonal knot energies, showing that the Minimum Distance Energy for polygons inscribed in a smooth knot converges to the Moebius Energy of the smooth knot as the polygons converge to the smooth knot. However, the polygons must converge in a ``nice'' way, and th…

Let $K$ be a knot type for which the quadratic term of the Conway polynomial is nontrivial, and let $γ: \mathbb{R}\to \mathbb{R}^3$ be an analytic $\mathbb{Z}$-periodic function with non-vanishing derivative which parameterizes a knot of type $K$ in space. We prove that there exists a sequence of numbers $0\leq t_1 < t…

A planar graph is inscribable if it is combinatorial equivalent to the skeleton of a polyhedra which is inscribed in a sphere. For an inscribable graph, in its combinatorial equivalent class, if we could always find polyhedra inscribed in any given convex surface which is sufficiently close to the sphere, then we call …

Extends sphere-rhomb inscribing to more directions.

Study examines Hilbert area of inscribed polygons in projective geometry.

Every curve can fit countless rhombuses.

Study of discrete Koenigs nets and their properties.

New bounds on inscribed triangles in arbitrary planar domains.

We prove that for every smooth Jordan curve $γ$, if $X$ is the set of all $r \in [0,1]$ so that there is an inscribed rectangle in $γ$ of aspect ratio $\tan(r\cdot π/4)$, then the Lebesgue measure of $X$ is at least $1/3$. To do this, we study sets of disjoint homologically nontrivial projective planes smoothly embedde…

Derives conformal parameters of curves using inscribed circular polygons.

Paper generalizes inscribed radius estimate to hyperbolic Einstein manifolds.

We prove a sharp estimate for the inscribed radius under certain fully nonlinear curvature flows. This estimate is asymptotically sharp on cylinders.

Square inscribed in a curve made of two graph functions.

Study on volumes of random inscribed polytopes in projective geometries.

Similar simplices can be inscribed in most smoothly embedded spheres.

We study convex polyhedra in $\mathbb{R}\mathbb{P}^3$ with all their vertices on a sphere. We do not require, in particular, that the polyhedra lie in the interior of the sphere, hence the term "weakly inscribed". Such polyhedra can be interpreted as ideal polyhedra, if we regard $\mathbb{R}\mathbb{P}^3$ as a combinati…

We study convex polyhedra in three-space that are inscribed in a quadric surface. Up to projective transformations, there are three such surfaces: the sphere, the hyperboloid, and the cylinder. Our main result is that a planar graph $Γ$ is realized as the $1$-skeleton of a polyhedron inscribed in the hyperboloid or cyl…

Sharp upper bounds on inscribed radius for metric spaces with convex boundary.

We prove that any cyclic quadrilateral can be inscribed in any closed convex $C^1$-curve. The smoothness condition is not required if the quadrilateral is a rectangle.

Floer homology applied to inscribing rectangles into curves.

We show that for every positive integer n there is a simple closed curve in the plane (which can be taken infinitely differentiable and convex) which has exactly n inscribed squares.

We consider a family of embedded, mean convex hypersurfaces which evolve by the mean curvature flow. It follows from general results of White that the inscribed radius at each point on the surface is at least $\frac{c}{H}$, where $c$ is a constant that depends only on the initial data. Andrews recently gave a new proof…

Square can fit inside curves close to smooth ones.

Square-like quadrilaterals inscribed in space curves proven for finite total curvature.

The abstract proves polygon inscriptions in curves with specific edge ratios.

The abstract shows how embeddings inscribe trapezoids or map three points to a line, proving nonexistence of certain maps.

We find bounds on the difference between the writhing number of a smooth curve, and the writhing number of a polygon inscribed within. The proof is based on an extension of Fuller's difference of writhe formula to the case of polygonal curves. The results establish error bounds useful in the computation of writhe.

In a recent paper, Brendle proved that the inscribed radius of closed embedded mean convex hypersurfaces moving by mean curvature flow is at least 1/((1+δ)H) at all points with H > C(δ,M_0). In this note, we give a shorter proof of Brendle's estimate, and of a more general result for alpha-Andrews flows, based on our r…

The paper proves that any smooth curve can have two similar inscribed rectangles.

Curves inscribe rectangles with positive area.

Continuous curves inscribe isosceles trapezoids in complex plane.

Paper studies inscribed sphere and lines through centers of Apollonius spheres in n dimensions.

We prove a transversality "lifting property" for compactified configuration spaces as an application of the multijet transversality theorem: the submanifold of configurations of points on an arbitrary submanifold of Euclidean space may be made transverse to any submanifold of the configuration space of points in Euclid…

We study complete minimal graphs in HxR, which take asymptotic boundary values plus and minus infinity on alternating sides of an ideal inscribed polygon &#915; in H. We give necessary and sufficient conditions on the "lenghts" of the sides of the polygon (and all inscribed polygons in &#915;) that ensure the existence…

First, we prove a special case of Knaster's problem, implying that each symmetric convex body in R^3 admits an inscribed cube. We deduce it from a theorem in equivariant topology, which says that there is no S_4-equivariant map from SO(3) to S^2, where S_4 acts on SO(3) as the rotation group of the cube and on S^2 as t…

We consider a family of embedded, mean convex hypersurfaces in a Riemannian manifold which evolve by the mean curvature flow. We show that, given any number $T>0$ and any $δ>0$, we can find a constant $C_0$ with the following property: if $t \in [0,T)$ and $p$ is a point on $M_t$ where the curvature is greater than $C_…

New Jordan Floer homology shows every curve inscribes every isosceles trapezoid.

Study compares manifolds with boundary under weighted Ricci curvature bounds.

The study connects polygon areas and projective structures in 3D space.

Consider a structured matrix factorization model where one factor is restricted to have its columns lying in the unit simplex. This simplex-structured matrix factorization (SSMF) model and the associated factorization techniques have spurred much interest in research topics over different areas, such as hyperspectral u…

We prove that, among all convex hyperbolic polygons with given angles, the perimeter is minimized by the unique polygon with an inscribed circle. The proof relies on work of J.-M.\ Schlenker.

A counterexample is given for the Knaster-like conjecture of Makeev for functions on $S^2$. Some particular cases of another conjecture of Makeev, on inscribing a quadrangle into a smooth simple closed curve, are solved positively.

Study finds Scherk type surfaces as extremals for zero-curvature minimal graphs.

We obtain sharp lower bounds on the radii of inscribed balls for strictly convex isoperimetric domains lying in a 2-dimensional Alexandrov metric space of curvature bounded below. We also characterize the case when such bounds are attained.