Related papers: Shortest closed curve to inspect a sphere

The circumference of a graph is the length of its longest cycles. Jackson established a conjecture of Bondy by showing that the circumference of a 3-connected cubic graph of order $n$ is $\Omega(n^{0.694})$. Bilinski {\it et al.} improved…

A simple closed curve in the Euclidean plane is said to have property C_n(R) if at each point we can inscribe a unique regular $n$-gon with edges length $R$. C_2(R) is equivalent to having constant diameter. We show that smooth curves…

The reductivity of a spherical curve is the minimal number of a local transformation called an inverse-half-twisted splice required to obtain a reducible spherical curve from the spherical curve. It is unknown if there exists a spherical…

We construct a family of Riemannian 3-spheres that cannot be "swept out" by short closed curves. More precisely, for each $L > 0$ we construct a Riemannian 3-sphere $M$ with diameter and volume less than 1, so that every 2-parameter family…

It is known that every closed curve of length \leq 4 in R^n (n>0) can be surrounded by a sphere of radius 1, and that this is the best bound. Letting S denote the circle of circumference 4, with the arc-length metric, we here express this…

Let S be a complete surface of constant curvature K = + 1 or -1, i.e. the sphere S^2 or the Lobachevskij plane L^2, and D a bounded convex subset of S. If S = S^2, assume also diameter (D) < pi/2. It is proved that the length of any…

A quasigeodesic is a curve on the surface of a convex polyhedron that has $\le \pi$ surface to each side at every point. In contrast, a geodesic has exactly $\pi$ to each side and so can never pass through a vertex, whereas quasigeodesics…

We prove that curves indicated in the title exist. This results answers to a question posed by A.G.Vitushkin about 30 years ago. We also discuss the minimal number of boundary components of a curve in the unit ball passing through the…

For a fixed radius $r$ and a point $o$ in the curve complex of a surface, we define the sphere of radius $r$ to be the induced subgraph on the set of vertices of distance $r$ from $o$. We show that these spheres are almost simply connected…

We study the lengths of curves passing through a fixed number of points on the boundary of a convex shape in the plane. We show that for any convex shape $K$, there exist four points on the boundary of $K$ such that the length of any curve…

We obtain an upper bound for the volume of the convex hull of a simple closed Frenet curve with exactly four vertices, i.e., four points of vanishing torsion, and lying on the boundary of its convex hull. Moreover, we show that the upper…

Let $M^n$ be a closed Riemannian manifold of dimension $n\geq 2$, with Ricci curvature $Ric \geq n-1$. We will show that any sphere of dimension $m$ in the space of closed loops on $M^n$ is homotopic to the sphere in the space of closed…

A cycle cover of a graph is a collection of cycles such that each edge of the graph is contained in at least one of the cycles. The length of a cycle cover is the sum of all cycle lengths in the cover. We prove that every bridgeless cubic…

A sphere packing of the three-dimensional Euclidean space is compact if it has only tetrahedral holes, that is, any local maximum of the distance to the spheres is at equal distance to exactly four spheres. This papers describes all the…

We find bounds for the length of the systole -- the shortest essential, non-peripheral closed curve -- for arithmetic punctured spheres with $n$ cusps, for $n=4$ through $n=12$, some of which were previously known due to Schmutz. This is…

In many singular metric spaces, the regularity of a shortest-length curve is unknown. Algebraic varieties, or more generally sets defined by finitely many polynomial or real analytic equalities or inequalities, all locally partition into…

The problem of finding small sets that block every line passing through a unit square was first considered by Mazurkiewicz in 1916. We call such a set {\em opaque} or a {\em barrier} for the square. The shortest known barrier has length…

How efficiently can a closed curve of unit length in $\mathbb{R}^d$ be covered by $k$ closed curves so as to minimize the maximum length of the $k$ curves? We show that the maximum length is at most $2k^{-1} - \frac{1}{4} k^{-4}$ for all…

We show that for any simple closed curve in the sphere at infinity of a Gromov hyperbolic 3-space with cocompact metric, there exist a properly embedded least area plane in the space spanning the given curve. This gives a positive answer to…

We develop a Morse-Lusternik-Schnirelmann theory for the distance between two points of a smoothly embedded circle in a complete Riemannian manifold. This theory suggests very naturally a definition of width that generalises the classical…