Pinwheels and bypasses

Mathematische Nachrichten, 1995

Proceedings of the American Mathematical Society, 1980

The American Mathematical Monthly, 2015

It is shown that if two planar convex n-gons are oppositely oriented, then the segments joining the corresponding vertices have a common transversal. A different formulation is also given in terms of two cars moving along two convex curves in opposite directions. Some possible analogues in 3-space are also considered, and an example is shown that the full analogue is not true in the space. * Supported by the European Union and co-funded by the European Social Fund under the project "Telemedicine-focused research activities on the field of Mathematics, Informatics and Medical sciences" of project number "TÁMOP-4.2.2.A-11/1/KONV-2012-0073" † Supported by NSF DMS-1265375 1 There was only one correct approach for the particular problem we are considering: it was by Péter Frenkel, who basically found the official solution.

Computational Geometry, 2003

Define a "slice" curve as the intersection of a plane with the surface of a polytope, i.e., a convex polyhedron in three dimensions. We prove that a slice curve develops on a plane without self-intersection. The key tool used is a generalization of Cauchy's arm lemma to permit nonconvex "openings" of a planar convex chain.

2003

In 1905, in his famous article “Sur les lignes géodésiques des surfaces convexes” [1] H. Poincaré posed a question on the existence of at least three geometrically different closed geodesics without self-intersections on any smooth convex two-dimensional surface (2-surface) M homeomorphic to the two-dimensional sphere (2-sphere) S2. Each such geodesic splits the surface M into two domains homeomorphic to 2-discs, and it is naturally to call it simple geodesic. In 1917, G.D. Birkhoff proved [2] the existence of at least one closed simple geodesic on M (in the late 20s he extended the result to the multidimensional case [3]). Nowadays this geodesic is called the (Birkhoff) “equator”. The presence of the Birkhoff equator serves as a basis for proving the existence of infinitely many (non-simple) closed geodesics on the considered surface. This very recent result has been established by V. Bangert [4] and J. Franks [5] in 1991-1992. In 1929 L.Luysternik and L.Shnirel’man gave a proof [6...

Publicationes Mathematicae Debrecen

Let C ⊂ E 2 be a convex body. The C-length of a segment is the ratio of its length to the half of the length of a longest parallel chord of C. By a relatively equilateral polygon inscribed in C we mean an inscribed convex polygon all of whose sides are of equal C-length. We prove that for every boundary point x of C and every integer k ≥ 3 there exists a relatively equilateral k-gon with vertex x inscribed in C. We discuss the C-length of sides of relatively equilateral k-gons inscribed in C and we reformulate this question in terms of packing C by k homothetical copies which touch the boundary of C. Let C be a convex body in Euclidean n-space E n. If pq is a longest chord of C in a direction l, we say that points p and q are opposite and we call pq a diametral chord of C in direction l. By the C-distance dist C (a, b) of a and b we mean the ratio of the Euclidean distance |ab| of a and b to the half of the Euclidean distance of end-points of a diametral chord of C parallel to ab (comp. [7]). We use here the term relative distance if there is no doubt about C. By the C-length of the segment ab we mean dist C (a, b). If C ⊂ E 2 , we define a C-equilateral k-gon as a convex k-gon all of whose sides have equal C-lengths. We also use the name relatively equilateral k-gon when C is fixed. Section 1 is of an auxiliary nature. It presents properties of the Cdistance, and especially properties of the C-distance of boundary points

Studia Scientiarum Mathematicarum Hungarica, 2005

If K ′ ⊂ K are convex bodies of the plane then the perimeter of K ′ is not greater than the perimeter of K. We obtain the following generalization of this fact. Let K be a convex compact body of the plane with the perimeter p and the diameter d and r > 1 be an integer. Let s be the smallest number such that for any curve of length greater than s contained in K there is a straight line intersecting the curve at least in r + 1 different points. Then s = rp/2 if r is even and s = (r − 1)p/2 + d if r is odd.

Arnold mathematical journal, 2024

Let A denote the cylinder R× S 1 or the band R× I , where I stands for the closed interval. We consider 2-moderate immersions of closed curves ("doodles") and compact surfaces ("blobs") in A, up to cobordisms that also are 2-moderate immersions in A × [0, 1] of surfaces and solids. By definition, the 2-moderate immersions of curves and surfaces do not have tangencies of order ≥ 3 to the fibers of the obvious projections . These bordisms come in different flavors: in particular, we consider one flavor based on regular embeddings of doodles and blobs in A. We compute the bordisms of regular embeddings and construct many invariants that distinguish between the bordisms of immersions and embeddings. In the case of oriented doodles on A = R × I , our computations of 2-moderate immersion bordisms OC imm moderate≤2 (A) are near complete: we show that they can be described by an exact sequence of abelian groups where OC emb moderate≤2 (A) ≈ Z × Z, the epimorphism Iρ counts different types of crossings of immersed doodles, and the kernel K contains the group (Z) ∞ whose generators are described explicitly.

Discrete & Computational Geometry, 1986

It is shown that if each pair of curves 7i, ~'j, i ~ j, intersect one another in at most two points, then the boundary of K =1.)~"= 1Ki contains at most max(2, 6m-12) intersection points of the curves y,, and this bound cannot be improved. As a corollary, we obtain a similar upper bound for the number of points of local nonconvexity in the union of rn Minkowski sums of planar convex sets. Following a basic approach suggested by Lozano Perez and Wesley, this can be applied to planning a collision-free translational motion of a convex polygon B amidst several (convex) polygonal obstacles A 1 ..... Am. Assuming that the number of corners of B is fixed, the algorithm presented here runs in time O(n log2n), where n is the total number of comers of the Ai's.

2001

Let R and B be point sets such that R ∪ B is in general position. We say that B is enclosed by R if there is a simple polygon P with vertex set R such that all the elements in B belong to the interior of P. In this paper we prove that if the vertices of the convex hull of R ∪ B belong to B, and |R| ≤ |Conv (R)| − 1 then B encloses R. The bound is tight. This improves on results of a previous paper in which it was proved that if |R| ≤ 56|Conv (B)| then B encloses R. To obtain our result we prove the next result which is interesting on its own right: Let P be a convex polygon with n vertices p 1 ,. .. , p n and S a set of m points contained in the interior of P , m ≤ n − 1. Then there is a convex decomposition {P 1 ,. .. , P n } of P such that all points from S lie on the boundaries of P 1 ,. .. , P n , and each P i contains a whole edge of P on its boundary.