53 research outputs found
Minimizers for an aggregation model with attractive-repulsive interaction
We solve explicitly a certain minimization problem for probability measures
involving an interaction energy that is repulsive at short distances and
attractive at large distances. We complement earlier works by showing that part
of the remaining parameter regime all minimizers are uniform distributions on a
surface of a sphere, thus showing concentration on a lower dimensional set. Our
method of proof uses convexity estimates on hypergeometric functions.Comment: 16 page
Polarization and Greedy Energy on the Sphere
We investigate the behavior of a greedy sequence on the sphere
defined so that at each step the point that minimizes the Riesz -energy is
added to the existing set of points. We show that for , the greedy
sequence achieves optimal second-order behavior for the Riesz -energy (up to
constants). In order to obtain this result, we prove that the second-order term
of the maximal polarization with Riesz -kernels is of order in the
same range . Furthermore, using the Stolarsky principle relating the
-discrepancy of a point set with the pairwise sum of distances (Riesz
energy with ), we also obtain a simple upper bound on the -spherical
cap discrepancy of the greedy sequence and give numerical examples that
indicate that the true discrepancy is much lower.Comment: 29 pages, 10 figure
Energy on spheres and discreteness of minimizing measures
In the present paper we study the minimization of energy integrals on the sphere with a focus on an interesting clustering phenomenon: for certain types of potentials, optimal measures are discrete or are supported on small sets. In particular, we prove that the support of any minimizer of the p-frame energy has empty interior whenever p is not an even integer. A similar effect is also demonstrated for energies with analytic potentials which are not positive definite. In addition, we establish the existence of discrete minimizers for a large class of energies, which includes energies with polynomial potentials
Optimal measures for p-frame energies on spheres
We provide new answers about the distribution of mass on spheres so as to minimize energies of pairwise interactions. We find optimal measures for the pp-frame energies, i.e., energies with the kernel given by the absolute value of the inner product raised to a positive power pp. Application of linear programming methods in the setting of projective spaces allows for describing the minimizing measures in full in several cases: we show optimality of tight designs and of the 600-cell for several ranges of pp in different dimensions. Our methods apply to a much broader class of potential functions, namely, those which are absolutely monotonic up to a particular order
A random line intersects in two probabilistically independent locations
We consider random lines in (random with respect to the
kinematic measure) and how they intersect . We prove that the
entry point and the exit point behave like independent uniformly distributed
random variables. This property is very rare; we prove that if is a bounded, convex domain with smooth boundary satisfying this
property (i.e., the intersection points with a random line are independent),
then and is a ball
Optimizers of three-point energies and nearly orthogonal sets
This paper is devoted to spherical measures and point configurations optimizing three-point energies. Our main goal is to extend the classic optimization problems based on pairs of distances between points to the context of three-point potentials. In particular, we study three-point analogues of the sphere packing problem and the optimization problem for p-frame energies based on three points. It turns out that both problems are inherently connected to the problem of nearly orthogonal sets by Erdős. As the outcome, we provide a new solution of the Erdős problem from the three-point packing perspective. We also show that the orthogonal basis uniquely minimizes the p-frame three-point energy when
Potential theory with multivariate kernels
In the present paper we develop the theory of minimization for energies with
multivariate kernels, i.e. energies, in which pairwise interactions are
replaced by interactions between triples or, more generally, -tuples of
particles. Such objects, which arise naturally in various fields, present
subtle differences and complications when compared to the classical two-input
case. We introduce appropriate analogues of conditionally positive definite
kernels, establish a series of relevant results in potential theory, explore
rotationally invariant energies on the sphere, and present a variety of
interesting examples, in particular, some optimization problems in
probabilistic geometry which are related to multivariate versions of the Riesz
energies.Comment: 23 page
- …