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There are three classical divergence measures exist in the literature on information theory and statistics. These are namely, Jeffryes-Kullback-Leiber J-divergence. Sibson-Burbea-Rao Jensen-Shannon divegernce and Taneja arithemtic-geometric…

For both triangulations of point sets and simple polygons, it is known that determining the flip distance between two triangulations is an NP-hard problem. To gain more insight into flips of triangulations and to characterize "where edges…

We present an algorithm of clustering of many-dimensional objects, where only the distances between objects are used. Centers of classes are found with the aid of neuron-like procedure with lateral inhibition. The result of clustering does…

We study the Coulomb chain where particles are restricted to one dimension and experience three-dimensional Coulomb interactions with their nearest and next-to-nearest neighbours. The distances between consecutive particles are treated as…

Let $K, D$ be $n$-dimensional convex bodes. Define the distance between $K$ and $D$ as $$ d(K,D) = \inf \{\lambda | T K \subset D+x \subset \lambda \cdot TK \}, $$ where the infimum is taken over all $x \in R^n$ and all invertible linear…

The inner product provides a conceptually and algorithmically simple method for calculating the comoving distance between two cosmological objects given their redshifts, right ascension and declination, and arbitrary constant curvature. The…

Non-Euclidean triangle centers can be described using homogeneous coordinates that are proportional to the generalized sines of the directed distances of a given center from the edges of the reference triangle. Identical homogeneous…

The geometric median, a notion of center for multivariate distributions, has gained recent attention in robust statistics and machine learning. Although conceptually distinct from the mean (i.e., expectation), we demonstrate that both are…

We show that the number of unit distances determined by n points in R^3 is O(n^{3/2}), slightly improving the bound of Clarkson et al. established in 1990. The new proof uses the recently introduced polynomial partitioning technique of Guth…

The bisector of two nonempty sets P and Q in a metric space is the set of all points with equal distance to P and to Q. A distance k-sector of P and Q, where k is an integer, is a (k-1)-tuple (C_1, C_2, ..., C_{k-1}) such that C_i is the…

Let $\{p_1, \ldots , p_n \} \subset {\Bbb{R}}^2$ be a separated point set, i.e., any two points have a distance at least $1$. Let $k \ge 1$ be an integer, and $1 \le t_1 < \ldots < t_k$ be real numbers. Let $\delta > 0$. Suppose for all $1…

A centrally symmetric $2d$-vertex combinatorial triangulation of the product of spheres $\S^i\times\S^{d-2-i}$ is constructed for all pairs of non-negative integers $i$ and $d$ with $0\leq i \leq d-2$. For the case of $i=d-2-i$, the…

Let $|A|$ denote the cardinality of a finite set $A$. For any real number $x$ define $t(x)=x$ if $x\geq1$ and 1 otherwise. For any finite sets $A,B$ let $\delta(A,B)$ $=$ $\log_{2}(t(|B\cap\bar{A}||A|))$. We define {This appears as…

Let ${\cal T}$ be a triangulation of a set ${\cal P}$ of $n$ points in the plane, and let $e$ be an edge shared by two triangles in ${\cal T}$ such that the quadrilateral $Q$ formed by these two triangles is convex. A {\em flip} of $e$ is…

We present a criterion when six points chosen on the sides of a triangle belong to the same conic. Using this tool we show how the two geometrical gems - celebrated Poncelet's theorem of projective geometry and incredible Morley's theorem…

A small triangulation of the sphere product can be found in lower dimensions by computer search and is known in few other cases: Klee and Novik constructed a centrally symmetric triangulation of $\mathbb{S}^i\times \mathbb{S}^{d-i-1}$ with…

In this short technical report, we define on the sample space R^D a distance between data points which depends on their correlation. We also derive an expression for the center of mass of a set of points with respect to this distance.

We show that absolute correlation distance satisfies a K-relaxed triangle inequality, with the best K = 2.

Consider $n$ $d$-dimensional vectors with iid entries from a lattice distribution $X$. We show that the probability that all distances between them are equal is asymptotically \[ C_n\cdot\frac{1}{d^{(m-1)/2}} \quad \text{for} \quad d \to…

The distances between flats of a Poisson $k$-flat process in the $d$-dimensional Euclidean space with $k<d/2$ are discussed. Continuing an approach originally due to Rolf Schneider, the number of pairs of flats having distance less than a…