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

Shapes from Echoes: Uniqueness from Point-to-Plane Distance Matrices

We study the problem of localizing a configuration of points and planes from the collection of point-to-plane distances. This problem models simultaneous localization and mapping from acoustic echoes as well as the notable "structure from sound" approach to microphone localization with unknown sources. In our earlier work we proposed computational methods for localization from point-to-plane distances and noted that such localization suffers from various ambiguities beyond the usual rigid body motions; in this paper we provide a complete characterization of uniqueness. We enumerate equivalence classes of configurations which lead to the same distance measurements as a function of the number of planes and points, and algebraically characterize the related transformations in both 2D and 3D. Here we only discuss uniqueness; computational tools and heuristics for practical localization from point-to-plane distances using sound will be addressed in a companion paper.Comment: 13 pages, 13 figure

On distinct distances in homogeneous sets in the Euclidean space

A homogeneous set of $n$ points in the $d$-dimensional Euclidean space determines at least $\Omega(n^{2d/(d^2+1)} / \log^{c(d)} n)$ distinct distances for a constant $c(d)>0$. In three-space, we slightly improve our general bound and show that a homogeneous set of $n$ points determines at least $\Omega(n^{.6091})$ distinct distances

Modelling the Galactic distribution of free electrons

In this paper we test 8 models of the free electron distribution in the Milky Way that have been published previously, and we introduce 4 additional models that explore the parameter space of possible models further. These new models consist of a simple exponential thick disk model, and updated versions of the models by Taylor & Cordes and Cordes & Lazio with more extended thick disks. The final model we introduce uses the observed H-alpha intensity as a proxy for the total electron column density, also known as the dispersion measure (DM). We use the latest available data sets of pulsars with accurate distances (through parallax measurements or association with globular clusters) to optimise the parameters in these models. In the process of fitting a new scale height for the thick disk in the model by Cordes & Lazio we discuss why this thick disk cannot be replaced by the thick disk that Gaensler et al. advocated in a recent paper. In the second part of our paper we test how well the different models can predict the DMs of these pulsars at known distances. Almost all models perform well, in that they predict DMs within a factor of 1.5-2 of the observed DMs for about 75% of the lines of sight. This is somewhat surprising since the models we tested range from very simple models that only contain a single exponential thick disk to very complex models like the model by Cordes & Lazio. We show that the model by Taylor & Cordes that we updated with a more extended thick disk consistently performs better than the other models we tested. Finally, we analyse which sightlines have DMs that prove difficult to predict by most models, which indicates the presence of local features in the ISM between us and the pulsar. (abridged)Comment: 16 pages, 10 figures, 5 tables. Accepted for publication in the Monthly Notices of the RAS by the Royal Astronomical Society and Blackwell Publishin