For each proper power of 4, n, we describe a simple explicit construction of a finite collection of pairwise disjoint open unit balls in R in which each ball touches more than 2 √ n others. A packing of balls in the Euclidean space is a finite or infinite collection of pairwise disjoint open unit balls in Rn. It is called a lattice packing if the centers of the balls form a lattice in Rn. The m...

We prove that there exists a norm in the plane under which no n-point set determines more than O(n log n log logn) 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).

We consider, and study with elementary calculus, the polyhedral norms $||x||_{(k)}=$ sum of $\mathit{k}$ largest among $|x_{i}|$'s. Besides their basic properties, we provide various expressions unit balls associated them determine all facets vertices these balls. do same dual norm $||.||_{(k)}^{\ast }$ $||.||_{(k)}$. The is motivated, other reasons, by necessity handling sparsity in some moder...

This study examines n-balls, n-simplices, and n-orthoplices in real dimensions using novel recurrence relations that remove the indefiniteness present known formulas. They show negative, integer dimensions, volumes of n-balls are zero if n is even, positive = ?4k ? 1, negative 3, for natural k. The surfaces n-cubes inscribed complex, wherein they associated with integral powers imaginary unit. ...

We provide a lower bound construction showing that the union of unit balls in R has quadratic complexity, even if they all contain the origin. This settles a conjecture of Sharir.

We extend Kahane-Khinchin type inequalities to the case p > −2. As an application we verify the slicing problem for the unit balls of finite-dimensional spaces that embed in Lp, p > −2.

The problem of construction of Barabanov norms for analysis of properties of the joint (generalized) spectral radius of matrix sets has been discussed in a number of publications. In [16, 17] the method of Barabanov norms was the key instrument in disproving the Lagarias-Wang Finiteness Conjecture. The related constructions were essentially based on the study of the geometrical properties of th...

We establish rigidity (or uniqueness) theorems for nc automorphisms which are natural extensions of clasical results of H. Cartan and are improvements of recent results. We apply our results to ncdomains consisting of unit balls of rectangular matrices.

To determine expertise-related differences in performance and movement variability during the execution of closed skill codified tasks, we quantitatively assessed the 3D hand movements of two groups of jugglers with different levels of expertise: six advanced (who could juggle up to 7 balls) and six intermediate jugglers (who could juggle at most with 5 balls). All participants performed three ...