Related papers: On Separation of Minimal Riesz Energy Points on Sp…

A balanced configuration of points on the sphere $S^2$ is a (finite) set of points which are in equilibrium if they act on each other according any force law dependent only on the distance between two points. The configuration is…

In this note we study estimates from below of the single radius spherical discrepancy in the setting of compact two-point homogeneous spaces. Namely, given a $d$-dimensional manifold $\mathcal M$ endowed with a distance $\rho$ so that…

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…

We develop a new approach to address some classical questions concerning the size and structure of integer distance sets. Our main result is that any integer distance set in the Euclidean plane is either very sparse or has all but an…

We determine explicit formulas for geodesics (in the Euclidean metric) in the configuration space of ordered pairs (x,x') of points in R^n which satisfy d(x,x')>=epsilon. We interpret this as two or three (depending on the parity of n)…

An $r^{\an}$-center of a compact body $\Om$ in an $n$ dimensional Euclidean space is a point that gives an extremal value of the regularized Riesz potential, which is the (Hadamard regularization of) integration on $\Om$ of the distance…

We study energy minimization for pair potentials among periodic sets in Euclidean spaces. We derive some sufficient conditions under which a point lattice locally minimizes the energy associated to a large class of potential functions. This…

Let $S$ be a set of points in $\mathbb{R}^2$ contained in a circle and $P$ an unrestricted point set in $\mathbb{R}^2$. We prove the number of distinct distances between points in $S$ and points in $P$ is at least…

The set of primitive vectors on large spheres in the euclidean space of dimension d>2 equidistribute when projected on the unit sphere. We consider here a refinement of this problem concerning the direction of the vector together with the…

We establish upper bounds for the size of two-distance sets in Euclidean space and spherical two-distance sets. The main recipe for obtaining upper bounds is the spectral method. We construct Seidel matrices to encode the distance relations…

It is shown that the supports of measures minimizing weakly repulsive energies on Riemannian manifolds with sectional curvature bounded below do not have concentration points. This extends the results of Bj\"orck and Carrillo, Figalli, and…

The Euclidean concentration inequality states that, among sets with fixed volume, balls have $r$-neighborhoods of minimal volume for every $r>0$. On an arbitrary set, the deviation of this volume growth from that of a ball is shown to…

Particles interacting through long-range attraction and short-range repulsion given by power-laws have been widely used to model physical and biological systems, and to predict or explain many of the patterns they display. Apart from rare…

The celebrated Dvoretzky theorem asserts that every $N$-dimensional convex body admits central sections of dimension $d = \Omega(\log N)$, which is nearly spherical. For many instances of convex bodies, typically unit balls with respect to…

In this paper we deal with the bounded critical points of a Riesz energy of attractive-repulsive type in dimension 1. Under suitable assumptions on the growth of the kernel in the origin, we are able to prove that they are continuous inside…

For each N>=c_d*n^{2d*(d+1)/(d+2)} we prove the existence of a spherical n-design on S^d consisting of N points, where c_d is a constant depending only on $d$.

Let P be an hyperplane in R^N, and denote by dH the Hausdorff distance. We show that for all positive radius r < 1 there is an epsilon > 0, such that if K is a Reifenberg-flat set in B(0; 1), a ball in R^N, that contains the origin, with…

We consider weak solutions to the incompressible Euler equations. It is shown that energy conservation holds in any Onsager critical class in which smooth functions are dense. The argument is independent of the specific critical regularity…

We consider a free energy functional defined on probability densities on the unit sphere $\mathbb{S}^d$, and investigate its global minimizers. The energy consists of two components: an entropy and a nonlocal interaction energy, which…

Let two distinct $N$-simplexes be given in an Euclidean or pseudo-Euclidean $N+1$ dimensional space as each is defined by the coordinates of its $N+1$ vertexes. We consider the two families of $N$-spheres passing through the vertexes of the…