Related papers: Mean Width of a Regular Simplex
The Straightness is a measure designed to characterize a pair of vertices in a spatial graph. It is defined as the ratio of the Euclidean distance to the graph distance between these vertices. It is often used as an average, for instance to…
The study of random surfaces, especially in the asymptotics of large genus, has been of increasing interest in recent years. Many geometrical questions have analogous formulations in the theory of random graphs with a large number of…
Similarly to the classic notion in $E^d$, a subset of a positive diameter below $\frac{\pi}{2}$ of a hemisphere of the sphere $S^d$ is called complete, provided adding any extra point increases its diameter. Complete sets are convex bodies…
The polygon $P$ is small if its diameter equals one. When $n=2^s$, it is still an open problem to find the maximum perimeter or the maximum width of a small $n$-gon. Motivated by Bingane's series of works, we improve the lower bounds for…
Given a Riemannian metric on a homotopy $n$-sphere, sweep it out by a continuous one-parameter family of closed curves starting and ending at point curves. Pull the sweepout tight by, in a continuous way, pulling each curve as tight as…
For every large enough $n$, we explicitly construct a body of constant width $2$ that has volume less than $0.9^n \text{Vol}(\mathbb{B}^{n}$), where $\mathbb{B}^{n}$ is the unit ball in $\mathbb{R}^{n}$. This answers a question of…
For a class of random band matrices of band width $W$, we prove regularity of the average spectral measure at scales $\epsilon \geq W^{-0.99}$, and find its asymptotics at these scales.
Recent results in quantization theory show that the mean-squared expected distortion can reach a rate of convergence of $\mathcal{O}(1/n)$, where $n$ is the sample size [see, e.g., IEEE Trans. Inform. Theory 60 (2014) 7279-7292 or Electron.…
When testing for the mean vector in a high dimensional setting, it is generally assumed that the observations are independently and identically distributed. However if the data are dependent, the existing test procedures fail to preserve…
A simplex in n dimensions is defined by the usual (n+1) linear inequality constraints in n dimensions. Here we consider simplexes which are bounded sets. The harmonic center has been defined earlier for polytopes in general. A relationship…
This thesis consists of five papers about reduced spherical convex bodies and in particular spherical bodies of constant width on the $d$-dimensional sphere $S^d$. In paper I we present some facts describing the shape of reduced bodies of…
The radius of convexity of two normalized Bessel functions of the first kind are determined in the case when the order is between $-2$ and $-1.$ Our methods include the minimum principle for harmonic functions, the Hadamard factorization of…
We investigate the geometric properties of the convex hull over $n$ successive positions of a planar random walk, with a symmetric continuous jump distribution. We derive the large $n$ asymptotic behavior of the mean perimeter. In addition,…
M. Gromov introduced the mean dimension for a continuous map in the late 1990's, which is an invariant under topological conjugacy. On the other hand, the notion of metric mean dimension for a dynamical system was introduced by…
This work introduces a complexity measure which addresses some conflicting issues between existing ones by using a new principle - measuring the average amount of symmetry broken by an object. It attributes low (although different)…
In the recent paper "On a formula for sets of constant width in 2D", Comm. Pure Appl. Anal. 18 (2019), 2117-2131, we gave a constructive formula for all 2d sets of constant width. Based on this result we derive here a formula for the…
Let $K$ be a convex body in $\mathbb{R}^d$ which slides freely in a ball. Let $K^{(n)}$ denote the intersection of $n$ closed half-spaces containing $K$ whose bounding hyperplanes are independent and identically distributed according to a…
We derive an explicit formula for the volume of a regular simplex in the hyperbolic space of any dimension.
This work describes a novel image analysis approach to characterize the uniformity of objects in agglomerates by using the propagation of normal wavefronts. The problem of width uniformity is discussed and its importance for the…
We derive bounds on the volume of an inclusion in a body in two or three dimensions when the conductivities of the inclusion and the surrounding body are complex and assumed to be known. The bounds are derived in terms of average values of…