Related papers: Navigating Around Convex Sets
We study the lengths of curves passing through a fixed number of points on the boundary of a convex shape in the plane. We show that for any convex shape $K$, there exist four points on the boundary of $K$ such that the length of any curve…
Extended object tracking considers the simultaneous estimation of the kinematic state and the shape parameters of a moving object based on a varying number of noisy detections. A main challenge in extended object tracking is the…
We consider two or more simple symmetric walks on some graphs, e.g. the real line, the plane or the two dimensional comb lattice, and investigate the properties of the distance among the walkers.
Distance covariance is a widely used statistical methodology for testing the dependency between two groups of variables. Despite the appealing properties of consistency and superior testing power, the testing results of distance covariance…
The geometric problem of estimating an unknown compact convex set from evaluations of its support function arises in a range of scientific and engineering applications. Traditional approaches typically rely on estimators that minimize the…
We briefly introduce several problems: (1) a generalization of the convex fair partition conjecture, (2) on non-trivial invariants among polyhedrons that can be formed from the same set of face polygons, (3) two questions on assembling…
Constraints on cosmological parameters are often distilled from sky surveys by fitting templates to summary statistics of the data that are motivated by a fiducial cosmological model. However, recent work has shown how to estimate the…
In the maritime sector, safe vessel navigation is of great importance, particularly in congested harbors and waterways. The focus of this work is to estimate the distance between an object of interest and potential obstacles using a…
We investigate the travel time in a navigation problem from a geometric perspective. The setting involves an open subset of the Euclidean plane, representing a lake perturbed by a symmetric wind flow proportional to the distance from the…
This article provides an overview on the statistical modeling of complex data as increasingly encountered in modern data analysis. It is argued that such data can often be described as elements of a metric space that satisfies certain…
Approximating distance is one of the key challenge in a facility location problem. Several algorithms have been proposed, however, none of them focused on estimating distance between two concave regions. In this work, we present an…
The traditional study of plane and space algebraic curves by looking at their tangent vectors, curvatures and torsions provides geometric, but unfortunately not sufficient information about individual curves in order to be able to…
We show that if K is a self-similar set in the plane with positive length, then the distance set of K has Hausdorff dimension one.
An $m$-distance set is a collection of points such that the distances between any two points have $m$ possible values. We use two different methods to construct large $m$-distance sets on the triangular lattices. One is to use the first m…
We prove some results concerning the boundary of a convex set in $\H^n$. This includes the convergence of curvature measures under Hausdorff convergence of the sets, the study of normal points, and, for convex surfaces, a generalized Gauss…
Let K be a d-dimensional convex body, and let $K^{(n)}$ be the intersection of n halfspaces containing $K$ whose bounding hyperplanes are independent and identically distributed. Under suitable distributional assumptions, we prove an…
We investigate methods for determining if a planar surface contains geometric deviations (e.g., protrusions, objects, divots, or cliffs) using only an instantaneous measurement from a miniature optical time-of-flight sensor. The key to our…
The work consists of solutions of metric problems for convex and finite subsets of geodesic spaces.
The geometry of jets of submanifolds is studied, with special interest in the relationship with the calculus of variations. A new intrinsic geometric formulation of the variational problem on jets of submanifolds is given. Working examples…
The covariogram g_K(x) of a convex body K \subseteq E^d is the function which associates to each x \in E^d the volume of the intersection of K with K+x. Matheron asked whether g_K determines K, up to translations and reflections in a point.…