The number of distinct distances from a vertex of a convex polygon

Erd\H{o}s conjectured in 1946 that every n-point set P in convex position in the plane contains a point that determines at least floor(n/2) distinct distances to the other points of P. The best known lower bound due to Dumitrescu (2006) is 13n/36 - O(1). In the present note, we slightly improve on this result to (13/36 + eps)n - O(1) for eps ~= 1/23000. Our main ingredient is an improved bound on the maximum number of isosceles triangles determined by P.Comment: 11 pages, 4 figure

The number of unit distances is almost linear for most norms

We prove that there exists a norm in the plane under which no n-point set determines more than O(n log n log log n) unit distances. Actually, most norms have this property, in the sense that their complement is a meager set in the metric space of all norms (with the metric given by the Hausdorff distance of the unit balls)

Examples, Counterexamples, and Enumeration Results for Foldings and Unfoldings between Polygons and Polytopes

We investigate how to make the surface of a convex polyhedron (a polytope) by folding up a polygon and gluing its perimeter shut, and the reverse process of cutting open a polytope and unfolding it to a polygon. We explore basic enumeration questions in both directions: Given a polygon, how many foldings are there? Given a polytope, how many unfoldings are there to simple polygons? Throughout we give special attention to convex polygons, and to regular polygons. We show that every convex polygon folds to an infinite number of distinct polytopes, but that their number of combinatorially distinct gluings is polynomial. There are, however, simple polygons with an exponential number of distinct gluings. In the reverse direction, we show that there are polytopes with an exponential number of distinct cuttings that lead to simple unfoldings. We establish necessary conditions for a polytope to have convex unfoldings, implying, for example, that among the Platonic solids, only the tetrahedron has a convex unfolding. We provide an inventory of the polytopes that may unfold to regular polygons, showing that, for n>6, there is essentially only one class of such polytopes.Comment: 54 pages, 33 figure

Algorithms for distance problems in planar complexes of global nonpositive curvature

CAT(0) metric spaces and hyperbolic spaces play an important role in combinatorial and geometric group theory. In this paper, we present efficient algorithms for distance problems in CAT(0) planar complexes. First of all, we present an algorithm for answering single-point distance queries in a CAT(0) planar complex. Namely, we show that for a CAT(0) planar complex K with n vertices, one can construct in O(n^2 log n) time a data structure D of size O(n^2) so that, given a point x in K, the shortest path gamma(x,y) between x and the query point y can be computed in linear time. Our second algorithm computes the convex hull of a finite set of points in a CAT(0) planar complex. This algorithm is based on Toussaint's algorithm for computing the convex hull of a finite set of points in a simple polygon and it constructs the convex hull of a set of k points in O(n^2 log n + nk log k) time, using a data structure of size O(n^2 + k)

Completely empty pyramids on integer lattices and two-dimensional faces of multidimensional continued fractions

In this paper we develop an integer-affine classification of three-dimensional multistory completely empty convex marked pyramids. We apply it to obtain the complete lists of compact two-dimensional faces of multidimensional continued fractions lying in planes with integer distances to the origin equal 2, 3, 4 ... The faces are considered up to the action of the group of integer-linear transformations. In conclusion we formulate some actual unsolved problems associated with the generalizations for n-dimensional faces and more complicated face configurations.Comment: Minor change