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, $n$-tuples of particles. Such objects, w
hich 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 multivaria
In this note, we give a short solution of the kissing number problem in dimension three.
A closed piecewise linear curve is called integral if it is comprised of unit intervals. Kenyons problem asks whether for every integral curve $gamma$ in $mathbb{R}^3$, there is a dome over $gamma$, i.e. whether $gamma$ is a boundary of a polyhedral
surface whose faces are equilateral triangles with unit edge lengths. First, we give an algebraic necessary condition when $gamma$ is a quadrilateral, thus giving a negative solution to Kenyons problem in full generality. We then prove that domes exist over a dense set of integral curves. Finally, we give an explicit construction of domes over all regular $n$-gons.
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 pr
ove 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.
We provide new answers about the placement of mass on spheres so as to minimize energies of pairwise interactions. We find optimal measures for the $p$-frame energies, i.e. energies with the kernel given by the absolute value of the inner product rai
sed to a positive power $p$. 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 $p$ in different dimensions. Our methods apply to a much broader class of potential functions, those which are absolutely monotonic up to a particular order as functions of the cosine of the geodesic distance. In addition, a preliminary numerical study is presented which suggests optimality of several other highly symmetric configurations and weighted designs in low dimensions. In one case we improve the best known lower bounds on a minimal sized weighted design in $mathbb{CP}^4$. All these results point to the discreteness of minimizing measures for the $p$-frame energy with $p$ not an even integer.
The paper is devoted to coverings by translative homothets and illuminations of convex bodies. For a given positive number $alpha$ and a convex body $B$, $g_{alpha}(B)$ is the infimum of $alpha$-powers of finitely many homothety coefficients less tha
n 1 such that there is a covering of $B$ by translative homothets with these coefficients. $h_{alpha}(B)$ is the minimal number of directions such that the boundary of $B$ can be illuminated by this number of directions except for a subset whose Hausdorff dimension is less than $alpha$. In this paper, we prove that $g_{alpha}(B)leq h_{alpha}(B)$, find upper and lower bounds for both numbers, and discuss several general conjectures. In particular, we show that $h_{alpha} (B) > 2^{d-alpha}$ for almost all $alpha$ and $d$ when $B$ is the $d$-dimensional cube, thus disproving the conjecture from Research Problems in Discrete Geometry by Brass, Moser, and Pach.
In this paper, we use the linear programming approach to find new upper bounds for the moments of isotropic measures. These bounds are then utilized for finding lower packing bounds and energy bounds for projective codes. We also show that the obtain
ed energy bounds are sharp for several infinite families of codes.
For a collection of $N$ unit vectors $mathbf{X}={x_i}_{i=1}^N$, define the $p$-frame energy of $mathbf{X}$ as the quantity $sum_{i eq j} |langle x_i,x_j rangle|^p$. In this paper, we connect the problem of minimizing this value to another optimizatio
n problem, so giving new lower bounds for such energies. In particular, for $p<2$, we prove that this energy is at least $2(N-d) p^{-frac p 2} (2-p)^{frac {p-2} 2}$ which is sharp for $dleq Nleq 2d$ and $p=1$. We prove that for $1leq m<d$, a repeated orthonormal basis construction of $N=d+m$ vectors minimizes the energy over an interval, $pin[1,p_m]$, and demonstrate an analogous result for all $N$ in the case $d=2$. Finally, in connection, we give conjectures on these and other energies.
A contact graph of a packing of closed balls is a graph with balls as vertices and pairs of tangent balls as edges. We prove that the average degree of the contact graph of a packing of balls (with possibly different radii) in $mathbb{R}^3$ is not gr
eater than $13.92$. We also find new upper bounds for the average degree of contact graphs in $mathbb{R}^4$ and $mathbb{R}^5$.
The set of points in a metric space is called an $s$-distance set if pairwise distances between these points admit only $s$ distinct values. Two-distance spherical sets with the set of scalar products ${alpha, -alpha}$, $alphain[0,1)$, are called equ
iangular. The problem of determining the maximum size of $s$-distance sets in various spaces has a long history in mathematics. We suggest a new method of bounding the size of an $s$-distance set in compact two-point homogeneous spaces via zonal spherical functions. This method allows us to prove that the maximum size of a spherical two-distance set in $mathbb{R}^n$, $ngeq 7$, is $frac{n(n+1)}2$ with possible exceptions for some $n=(2k+1)^2-3$, $k in mathbb{N}$. We also prove the universal upper bound $sim frac 2 3 n a^2$ for equiangular sets with $alpha=frac 1 a$ and, employing this bound, prove a new upper bound on the size of equiangular sets in all dimensions. Finally, we classify all equiangular sets reaching this new bound.
