Voronoi diagrams in higher dimensions under certain polyhedral distance functions

The bisector systems of convex distance functions in 3-space are investigated and it is shown that there is a substantial difference to the Euclidean metric which cannot be observed in 2-space. This disproves the general belief that Voronoi diagrams in convex distance functions are, in any dimension, analogous to Euclidean Voronoi diagrams. The fact is that more spheres than one can pass through four points in general position. In the L4-metric, there exist quadrupels of points that lie on the surface of three L4-spheres. Moreover, for each n ≥ 0 one can construct a smooth and symmetric convex distance function d and four points that are contained in the surface of exactly 2n+1+ d-spheres, and this number does not decrease if the four points are disturbed independently within 3-dimensional neighborhoods. This result implies that there is no general upper bound to the complexity of the Voronoi diagram of four sites based on a convex distance function in 3-space.

In this paper we present some efficient and a few optimal algorithms to compute the radius, diameter, incenter, circumcenter, width and k-dimensional enclosing cylinder for convex polyhedral and convex polyhedral offset distance functions in plane and in \(\mathfrak {R}^d\). The radius, incenter and circumcenter of a convex polygon can be optimally computed in linear time, i.e. \(O(n+m)\), in \(\mathfrak {R}^2\) and in time \(O(n+m+|R|)\), if the incenter or circumcenter is additionally constrained in a convex polygon R, where n is the size of input polygon and m is the size of polygon in convex polygonal or convex polygonal offset distance functions. In \(\mathfrak {R}^d\), the radius, incenter and circumcenter of a convex polyhedron as well as of a set of convex polyhedra, unconstrained or constrained, can be computed in O(nm) and \(O(nm+|R|)\) time respectively, for convex constraint polyhedron R. The diameter of a convex polygon in plane can be computed in \(O(n+m)\) time. The diameter of a polyhedron in \(\mathfrak {R}^d\) can be computed in O(nm) time by our methods. The width of a convex polyhedron can be computed in \(O(n + m)\) time and O(nm) time, for \(\mathfrak {R}^2\) and \(\mathfrak {R}^d\), respectively. The diameter and width of a set of convex polyhedra can be computed in O(nm)-time. We also show how the k-dimensional minimum enclosing cylinder can be computed in \(O(n^{d-k+1}nm^2)\) in \(\mathfrak {R}^d\) and in \(O(n^{d}m(n+m))\) in \(\mathfrak {R}^d\) for \(k=1\). The k-dimensional minimum enclosing cylinder for a set of convex polyhedra in \(\mathfrak {R}^d\) can be computed in \(O(n^{d-k+1}nm^2)\)-time. The diameter and width problems can also be solved for a set of points in \(\mathfrak {R}^d\), either in time O(nm) or in time O(mT(n)), where T(n) is the time complexity of the best convex hull computation algorithm for the given set of points. We also compute the minimum stabbing sphere for a set of convex polyhedra, unconstrained or constrained, for the above mentioned distance functions in O(nm)-time or \(O(nm+|R|)\)-time respectively for convex constraint polyhedron R.

Separating collections of points in Euclidean spaces

This paper starts the investigation of a constrained version of the point-set embed-dability problem. Let G = (V,E) be a planar graph with n vertices, G′ = (V′,E′) a subgraph of G, and S a set of n distinct points in the plane. We study the problem of computing a point-set embedding of G on S subject to the constraint that G′ is drawn with straight-line edges. Different drawing algorithms are presented that guarantee small curve complexity of the resulting drawing, i.e. a small number of bends per edge. It is proved that: • If G′ is an outerplanar graph and S is any set of points in convex position, a point-set embedding of G on S can be computed such that the edges of E\E′ have at most 4 bends each. • If S is any set of points in general position and G′ is a face of G or if it is a simple path, the curve complexity of the edges of E\E′ is at most 8. • If S is in general position and G′ is a set of k disjoint paths, the curve complexity of the edges of E \ E′ is O(2k).

In this paper we give a lower bound for the Erd\H os–Szekeres number in higher dimensions. Namely, in two different ways we construct, for every $n>d\ge 2$, a configuration of $n$ points in general position in $\R^d$ containing at most $c_d(\log n)^{d-1}$ points in convex position. (Points in $\R^d$ are in convex position if none of them lies in the convex hull of the others.)

The classical Cover results on linear separability of points in Rd are a milestone in neural network theory. Nevertheless they are not valid for digital input networks because in this case the points are not, in general position. vertices ofa d-dimensional hypercube. I show here that for large d all Cover findings can be extended to this case. It is also shown that for n < O((d + I)*) the number of linear separations of n random hypercube vertices tends to that of n points in general position.

Three polynomials are defined for given sets $S$ of $n$ points in general position in the plane: The Voronoi polynomial with coefficients the numbers of vertices of the order-$k$ Voronoi diagrams of $S$, the circle polynomial with coefficients the numbers of circles through three points of $S$ enclosing $k$ points of $S$, and the $E_{\leq k}$ polynomial with coefficients the numbers of (at most $k$)-edges of $S$. We present several formulas for the rectilinear crossing number of $S$ in terms of these polynomials and their roots. We also prove that the roots of the Voronoi polynomial lie on the unit circle if, and only if, $S$ is in convex position. Further, we present bounds on the location of the roots of these polynomials.

Abstract. Erdös asked whether every large enough set of points in general position in the plane contains six points that form a convex 6-gon without any points from the set in its interior. In this note we show how a set of 29 points was found that contains no empty convex 6-gon. To this end a fast incremental algorithm for finding such 6-gons was designed and implemented and a heuristic search approach was used to find promising sets. The experiments also led to two observations that might be useful in proving that large sets always contain an empty convex 6-gon.

Obstacle representations of graphs have been investigated quite intensely over the last few years. We focus on graphs that can be represented by a single obstacle. Given a (topologically open) non-self-intersecting polygon C and a finite set P of points in general position in the complement of C, the visibility graph \(G_C(P)\) has a vertex for each point in P and an edge pq for any two points p and q in P that can see each other, that is, \(\overline{pq} \cap C=\emptyset \). We draw \(G_C(P)\) straight-line and call this a visibility drawing. Given a graph G, we want to compute an obstacle representation of G, that is, an obstacle C and a set of points P such that \(G=G_C(P)\). The complexity of this problem is open, even when the points are exactly the vertices of a simple polygon and the obstacle is the complement of the polygon—the simple-polygon visibility graph problem.

On embedding triconnected cubic graphs on point sets

Lower bounds on the maximum number of non-crossing acyclic graphs

Empty rainbow triangles in k-colored point sets

Let S be a finite set of n+3 points in general position in the plane, with 3 extreme points and n interior points. We consider triangulations drawn uniformly at random from the set of all triangulations of S, and investigate the expected number, vi, of interior points of degree i in such a triangulation. We provide bounds that are linear in n on these numbers. In particular, n/43≤v3≤ (2n+3)/5.Moreover, we relate these results to the question about the maximum and minimum possible number of triangulations in such a set S, and show that the number of triangulations of any set of n points in the plane is at most 43n, thereby improving on a previous bound by Santos and Seidel.

Kontsevich's formula is a recursion that calculates the number of rational degree $d$ curves in $\mathbb{P}_{\mathbb{C}}^2$ passing through $3d-1$ points in general position. Kontsevich proved it by considering curves that satisfy extra conditions besides the given point conditions. These crucial extra conditions are two line conditions and a condition called cross-ratio.&#x0D; This paper addresses the question whether there is a general Kontsevich's formula which holds for more than one cross-ratio. Using tropical geometry, we obtain such a recursive formula. For that, we use a correspondence theorem of Tyomkin that relates the algebro-geometric numbers in question to tropical ones. It turns out that the general tropical Kontsevich's formula we obtain is capable of not only computing the algebro-geometric numbers we are looking for, but also of computing further tropical numbers for which there is no correspondence theorem yet.&#x0D; We show that our recursive general Kontsevich's formula implies the original Kontsevich's formula and that the initial values are the numbers Kontsevich's fomula provides and purely combinatorial numbers, so-called cross-ratio multiplicities.

A linear system of plane curves satisfying multiplicity conditions at points in general position is called special if the dimension is larger than the expected dimension. A (−1) curve is an irreducible curve with self intersection −1 and genus zero. The Segre–Harbourne–Hirschowitz Conjecture states that if a linear system is special then a multiple of some fixed (−1) curve is contained in every curve of the linear system. This conjecture is proven for linear systems with multiplicity four at all but one of the points.