Related papers: Distance-squared mappings
We give a complete characterization of closed sets $F \subset \mathbb{R}^2$ whose distance function $d_F:= \mathrm{dist}(\cdot,F)$ is DC (i.e., is the difference of two convex functions on $\mathbb{R}^2$). Using this characterization, a…
We first introduce a class of divergence measures between power spectral density matrices. These are derived by comparing the suitability of different models in the context of optimal prediction. Distances between "infinitesimally close"…
These notes are designed for those who either plan to work in differential geometry, or at least want to have a good reason not to do it. We discuss smooth curves and surfaces -- the main gate to differential geometry. We focus on the…
We study a function, which is a weighted sum of the squares of the distances of an arbitrary point to the sidelines of a triangle. The given weights, considered as barycentric coordinates, determine a point $M$. We prove that the function…
We analyse the geometric properties of the high derivatives of the distance function from a submanifold of the Euclidean space. In particular, we show some relations with the second fundamental form and its covariant derivatives of…
In this paper we systematically study various properties of the distance graph in ${\Bbb F}_q^d$, the $d$-dimensional vector space over the finite field ${\Bbb F}_q$ with $q$ elements. In the process we compute the diameter of distance…
Distance functions between points in a domain are sometimes used to automatically plan a gradient-descent path towards a given target point in the domain, avoiding obstacles that may be present. A key requirement from such distance…
A mathematical smooth function means that the function has continuous derivatives to a certain degree C(k). We call it a k-smooth function or a smooth function if k can grow infinitively. Based on quantum physics, there is no such smooth…
In this paper, we propose the Fourier Discrepancy Function, a new discrepancy to compare discrete probability measures. We show that this discrepancy takes into account the geometry of the underlying space. We prove that the Fourier…
In the paper, some concepts of modern differential geometry are used as a basis to develop an invariant theory of mechanical systems, including systems with gyroscopic forces. An interpretation of systems with gyroscopic forces in the form…
Distance transformation is an image processing technique used for many different applications. Related to a binary image, the general idea is to determine the distance of all background points to the nearest object point (or vice versa). In…
We study the regularity of the distance function to the boundary of a domain in $\mathbb{R}^2$, with respect to some asymmetric norms. We allow the boundary of the domain to have corners. We obtain an explicit formula for the second…
Let $(X,d)$ be a compact metric space. We consider the behavior of probability measures $\mu$ with the property that $$ \int_{X} d(x, y) d\mu(y) \qquad \mbox{is independent of}~x \in X.$$ It appears that such measures, when they exist,…
This research work aims to explore the distortions in distance in equidistant cylindrical projection. The horizontal bending that occurs in the projection process can be assessed by performing a geometric analysis using Tissot's…
Statistical distances, divergences, and similar quantities have a large history and play a fundamental role in statistics, machine learning and associated scientific disciplines. However, within the statistical literature, this extensive…
Metric graphs are meaningful objects for modeling complex structures that arise in many real-world applications, such as road networks, river systems, earthquake faults, blood vessels, and filamentary structures in galaxies. To study metric…
For the distance function from any closed subset of any complete Finsler manifold, we prove that the singular set is equal to a countable union of delta-convex hypersurfaces up to an exceptional set of codimension two. In addition, in…
A classical approach for surface classification is to find a compact algebraic representation for each surface that would be similar for objects within the same class and preserve dissimilarities between classes. We introduce Self…
A distance mean function measures the average distance of points from the elements of a given set of points (focal set) in the space. The level sets of a distance mean function are called generalized conics. In case of infinite focal points…
Hypergeometric functions and their generalizations play an important r\^{o}les in diverse applications. Many authors have been established generalizations of hypergeometric functions by a number ways. In this paper, we aim at establishing…