Helly-type theorems for homothets of planar convex curves

Helly's theorem implies that if is a finite collection of (positive) homothets of a planar convex body , any three having non-empty intersection, then has non-empty intersection. We show that for collections of homothets (including translates) of the boundary , if any four curves in have non-empty intersection, then has non-empty intersection. We prove the following dual version: If any four points of a finite set in the plane can be covered by a translate [homothet] of , then can be covered by a translate [homothet] of . These results are best possible in general.

     

On the minimal convex annulus of a planar convex body

In this paper we introduce the notion of a minimal convex annulusK (C) of a convex bodyC, generalizing the concept of a minimal circular annulus. Then we prove the existence — as for the minimal circular annulus — of a Radon partition of the set of contact points of the boundaries ofK (C) andC. Subsequently, the uniqueness ofK (C) is shown. Finally, it is proven that, for typicalC, the boundary ofC has precisely two points in common with each component of the boundary ofK (C).     

On convex decompositions of a planar point set

Let P be a planar point set in general position. Neumann-Lara et al. showed that there is a convex decomposition of P with at most elements. In this paper, we improve this upper bound to .     

On a class of locally convex closed curves

Summary For a certain type of locally convex plane curves some relations are shown between the numbers of double points and double tangents and the order and class of the curve. It is proved that the curve has an equal number of double points and double tangents and at least v-1 where 2v denotes the order (and class) of the curve. At last it is shown how a curve with more that v-1 double points (tangents) may be deformed into a curve with the minimum number. To Enrico Bompiani on his scientific Jubilee     

Evolving convex curves

We consider the behaviour of convex curves undergoing curvature-driven motion. In particular we describe the long-term behaviour of solutions and properties of limiting shapes, and prove existence of unique solutions from singular or non-strictly convex initial curves, with sharp regularity estimates. Received May 21, 1997 / Accepted December 5, 1997     

Typical convex curves on convex surfaces

In the sense of Baire categories, most convex curves on a smooth twodimensional closed convex surface are smooth. Moreover, if the set of all closed geodesics has empty interior in the space of all convex curves, then most convex curves are strictly convex.This paper was written during the author's visit at Western Washington University, whose substantial support is acknowledged.     

On geometric interpolation by planar parametric polynomial curves

In this paper the problem of geometric interpolation of planar data by parametric polynomial curves is revisited. The conjecture that a parametric polynomial curve of degree can interpolate given points in is confirmed for under certain natural restrictions. This conclusion also implies the optimal asymptotic approximation order. More generally, the optimal order can be achieved as soon as the interpolating curve exists.

     

On empty convex polygons in a planar point set

Let P be a set of n points in general position in the plane. Let X_k(P) denote the number of empty convex k-gons determined by P. We derive, using elementary proof techniques, several equalities and inequalities involving the quantities X_k(P) and several related quantities. Most of these equalities and inequalities are new, except for a few that have been proved earlier using a considerably more complex machinery from matroid and polytope theory, and algebraic topology. Some of these relationships are also extended to higher dimensions. We present several implications of these relationships, and discuss their connection with several long-standing open problems, the most notorious of which is the existence of an empty convex hexagon in any point set with sufficiently many points.     

On the diameter of planar curves and Fourier coefficients

Summary LetF denote a complex valued function on , which is absolutely continuous and periodic with a period 2. Its complex Fourier coefficientsc _n satisfy the inequality 2|c _n |d,n0, whered denotes the diameter of the closed planar curve defined byF. In this paper those functionsF are considered, which give equality for a specificn.

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Zusammenfassung Es seiF eine komplexwertige Funktion auf , die absolut stetig und periodisch ist mit einer Periode 2. Sie definiert eine geschlossene ebene Kurve. Zwischen ihrem Durchmesserd und den komplexen Fourierkoeffizientenc _n besteht die Ungleichung 2|c _n |d fürn0. Es werden jene FunktionenF betrachtet, für die in dieser Ungleichung, für ein gegebenesn, Gleichheit besteht.

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Dedicated to my colleague for many years, Professor Eduard Stiefel

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This research was supported in part by the National Research Council of Canada A-7339     

On the development of closed convex curves on 3-polytopes

It is shown that a closed convex polygonal curve on the surface of a 3-polytope develops in the plane to a simple path: it does not self-intersect.     

The singular evolutoids set and the extended evolutoids front

Aequationes mathematicae - In this paper we introduce the notion of the singular evolutoid set which is the set of all singular points of all evolutoids of a fixed smooth planar curve with at most...