Abstract For many extremal configurations of points on a sphere, the linear programming approach can be used to show their optimality. In this paper we establish general framework for showing stability such and use prove two spherical codes formed by minimal vectors lattice $$E_8$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>8</mml:mn> </mml:msub> </mml:math> Leech lattice.

We provide new answers about the distribution of mass on spheres so as to minimize energies pairwise interactions. find optimal measures for p -frame energies, i.e., with kernel given by absolute value inner product raised a positive power . Application linear programming methods in setting projective spaces allows describing minimizing full several cases: we show optimality tight designs and 600-cell ranges different dimensions. Our apply much broader class potential functions, namely,...

In this paper, we use the linear programming approach to find new upper bounds for moments of isotropic measures. These are then utilized finding lower packing and energy projective codes. We also show that obtained sharp several infinite families

For a collection of $N$ unit vectors $\mathbf{X}=\{x_i\}_{i=1}^N$, define the $p$-frame energy $\mathbf{X}$ as quantity $\sum_{i\neq j} |\langle x_i,x_j \rangle|^p$. In this paper, we connect problem minimizing value to another optimization problem, so giving new lower bounds for such energies. particular, $p<2$, prove that is at least $2(N-d) p^{-\frac p 2} (2-p)^{\frac {p-2} 2}$ which sharp $d\leq N\leq 2d$ and $p=1$. We $1\leq m<d$, repeated orthonormal basis construction $N=d+m$...

The set of points in a metric space is called an $s$-distance if pairwise distances between these admit only $s$ distinct values. Two-distance spherical sets with the scalar products $\{\alpha, -\alpha\}$, $\alpha\in[0,1)$, are equiangular. problem determining maximum size various spaces has long history mathematics. We suggest new method bounding compact two-point homogeneous via zonal functions. This allows us to prove that two-distance $\mathbb{R}^n$, $n\geq 7$, $\frac{n(n+1)}2$ possible...

Abstract We note that the recent polynomial proofs of spherical and complex plank covering problems by Zhao Ortega-Moreno give some general information on zeros real polynomials restricted to unit sphere. As a corollary these results, we establish several generalizations celebrated Bang theorem. prove tight analog theorem for Euclidean ball an even stronger version projective space. Specifically, ball, show every nonzero $d$-variate $P$ degree $n$, there exists point in $d$-dimensional at...

We address the maximum size of binary codes and constant weight with few distances. Previous works established a number bounds for these quantities as well exact values range small code lengths. As our main results, we determine maximal two distances all lengths \(n\ge 6\) \(2\), \(3\), \(4\) several but lengths.Mathematics Subject Classifications: 52C10, 05D05, 94B65Keywords: Johnson space, Erdös-Ko-Rado, Delsarte inequalities

This paper is devoted to spherical measures and point configurations optimizing three-point energies. Our main goal extend the classic optimization problems based on pairs of distances between points context potentials. In particular, we study analogues sphere packing problem for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation>...

In this note we introduce a pseudometric on closed convex planar curves based distances between normal lines and show its basic properties. Then use to give shorter proof of the theorem by Pinchasi that sum perimeters 𝑘 bodies with disjoint interiors contained in body perimeter 𝑝 diameter 𝑑 is not greater than + 2(𝑘 − 1)𝑑.

We provide new answers about the placement of mass on spheres so as to minimize energies pairwise interactions. find optimal measures for $p$-frame energies, i.e. with kernel given by absolute value inner product raised a positive power $p$. Application linear programming methods in setting projective spaces allows describing minimizing full several cases: we show optimality tight designs and $600$-cell ranges $p$ different dimensions. Our apply much broader class potential functions, those...

We study densities of functionals over uniformly bounded triangulations a Delaunay set vertices, and prove that the minimum is attained for triangulation if this case finite sets.