Chebyshev systems and zeros of a function on a convex curve

龙曲线的凸包

Convex Hulls of Dragon Curves

本文研究了龙曲线的凸包性质，证明了每条龙曲线都有多边形凸包，并对其凸包进行了完整刻画。

On convex closed planar curves as equidistant sets

The equidistant set of two nonempty subsets $K$ and $L$ in the Euclidean plane is a set all of whose points have the same distance from $K$ and $L$. Since the classical conics can be also given in this way, equidistant sets can be considered as a kind of their generalizations: $K$ and $L$ are called the focal sets. In their paper \cite{PS} the authors posed the problem of the characterization of closed subsets in the Euclidean plane that can be realized as the equidistant set of two connected disjoint closed sets. We prove that any convex closed planar curve can be given as an equidistant set,…

Cutting convex curves

We show that for any two convex curves $C_1$ and $C_2$ in $\mathbb R^d$ parametrized by $[0,1]$ with opposite orientations, there exists a hyperplane $H$ with the following property: For any $t\in [0,1]$ the points $C_1(t)$ and $C_2(t)$ are never in the same open halfspace bounded by $H$. This will be deduced from a more general result on equipartitions of ordered point sets by hyperplanes.

凸 Peano 曲线确实存在

The Convex Peano Curve Does Exist

本文推广 Peano 的空间填充曲线构造，证明对任意 R² 中的凸有界闭集 T，都存在连续满射 [0,1]→T，并保证每个区间的像均为凸集。

On the Lengths of Curves Passing through Boundary Points of a Planar Convex Shape

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 passing through these points is at least half of the perimeter of $K$. It is also shown that the same statement does not remain valid with the additional constraint that the points are extreme points of $K$. Moreover, the factor $\frac12$ cannot be achieved with any fixed number of extreme points. We conclude the paper with few other inequalities related to …

The DNA Inequality in Non-Convex Regions

A simple plane closed curve $Γ$ satisfies the DNA Inequality if the average curvature of any closed curve contained inside $Γ$ exceeds the average curvature of $Γ$. In 1997 Lagarias and Richardson proved that all convex curves satisfy the DNA Inequality and asked whether this is true for any non-convex curve. They conjectured that the DNA Inequality holds for certain L-shaped curves. In this paper, we disprove this conjecture for all L-Shapes and construct a large class of non-convex curves for which the DNA Inequality holds. We also give a polynomial-time procedure for determining whether any…

Any cyclic quadrilateral can be inscribed in any closed convex smooth curve

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.

Properties of a curve whose convex hull covers a given convex body

In this note, we prove the following inequality for the norm of a convex body $K$ in $\mathbb{R}^n$, $n\geq 2$: $N(K) \leq \frac{π^{\frac{n-1}{2}}}{2 Γ\left(\frac{n+1}{2}\right)}\cdot \operatorname{length} (γ) + \frac{π^{\frac{n}{2}-1}}{Γ\left(\frac{n}{2}\right)} \cdot \operatorname{diam}(K)$, where $\operatorname{diam}(K)$ is the diameter of $K$, $γ$ is any curve in $\mathbb{R}^n$ whose convex hull covers $K$, and $Γ$ is the gamma function. If in addition $K$ has constant width $Θ$, then we get the inequality $\operatorname{length} (γ) \geq \frac{2(π-1)Γ\left(\frac{n+1}{2}\right)}{\sqrtπ\,Γ\l…

On a number of rational points on a convex curve

Let $γ$ be a bounded convex curve on a plane. Then $\sharp (γ\cap (\Z/n)^2)=o(n^{2/3})$. It streghtens the classical result of Jarn\'ık (an upper estimate $O(n^{2/3})$) and disproves a conjecture of Vershik on existence of the so-called {\it universal Jarn\'ık curve}.

A convex cover for closed unit curves has area at least 0.0975

We combine geometric methods with numerical box search algorithm to show that the minimal area of a convex set on the plane which can cover every closed plane curve of unit length is at least 0.0975. This improves the best previous lower bound of 0.096694. In fact, we show that the minimal area of convex hull of circle, equilateral triangle, and rectangle of perimeter $1$ is between 0.0975 and 0.09763.

Convex curves and a Poisson imitation of lattices

We solve a randomized version of the following open question: is there a strictly convex, bounded curve γin the plane such that the number of rational points on γ, with denominator $n$, approaches infinity with $n$? Although this natural problem appears to be out of reach using current methods, we consider a probabilistic analogue using a spatial Poisson-process that simulates the refined rational lattice $\frac{1}{d} Z^2$, which we call $M_d$, for each natural number $d$. The main result here is that with probability 1 there exists a strictly convex, bounded curve γsuch that the number of spa…