Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.34845

Optimization hierarchies for extremal geometry through complete positivity

Bram Bekker
View original source ↗Read source document ↗Export metadata ↓

Abstract

Completely positive functions are an extension of completely positive matrices. They are known to characterize maximal spherical codes and maximum-density distance-avoiding subsets of $\mathbb{R}^n$ and certain compact metric spaces. This thesis expands this framework to related classes of problems in finite measure spaces and to the sphere-packing problem. For the latter, this is sharpened to show that the optimal sphere-packing density can be approximated using Schwartz functions. Converging hierarchies of semidefinite programming bounds on the size of optimal spherical codes are known, based on approximations of completely positive functions and the Lovász theta number of a graph. This thesis extends these hierarchies to distance-avoiding sets and similar problems and to the sphere-packing problem, and proves their convergence to the maximum density. For distance-avoiding sets, additional hierarchies, such as the moment hierarchy, are introduced and shown to be stronger than the completely positive hierarchy, hence they also converge. Related hierarchies for compact packing problems are also investigate. These bounds are implemented for Witsenhausen's problem, which asks for the maximum fraction $α_n$ of the $n$-dimensional unit sphere that is coverable by a set avoiding orthogonal pairs; and for the $t$-almost-equiangular-set problem: finding the maximum size $α(n,t)$ of a subset of the $n$-dimensional unit sphere in which every triple contains a pair with inner product $t \in [-1,1)$. An analytic solution to this bound yields an enumeration of optimal constructions for $n = 2$ and $3$ when $t \geq 0$.

Explore related subjects

math.COmath.MGmath.OC

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Bram Bekker. 2026-09-28. Optimization hierarchies for extremal geometry through complete positivity. https://arxiv.org/abs/2609.34845

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The Rainbow Saturation Number of Cycles

An edge-colored graph $(G,\mathcal C)$ is $F$-rainbow saturated if it contains no rainbow copy of $F$, but adding any nonedge in any prescribed color creates one. The rainbow saturation number $\operatorname{rsat}(n,F)$ is the minimum number of edges in such a graph of order $n$. In this paper we study $\operatorname{rsat}(n,C_r)$. We determine the value for $C_4$ exactly, and establish bounds for every fixed $C_r$ with $r\geq5$.

math.CO↗
arXiv2026

Moreira's Theorem for Image Partition Regular Matrices

The famous Hindman conjecture says that for any finite coloring of natural numbers, there exists a monochromatic copy of the form $\{x,x+y,xy\}.$ In a celebrated article, Moreira gave a partial answer to this conjecture by showing that every finite coloring of the natural numbers contains a monochromatic configuration of the form $\{x, x+y, xy\}$. In this article we prove matrix versions (both finite and infinite) of Moreira's theorem. A matrix $A$ is said to be an image partition regular matrix if for any finite coloring of naturals, there exists a monochromatic image of $A,$ i.e. there exists a vector $\vec X$ such that all the entries of $A\vec X$ are monochromatic. From a recent paper of Bowen, one can derive the finite matrix version of the Moreira theorem: if $A$ and $B$ are two finite image partition regular matrices of the same order, then under any finite coloring of $\mathbb{N}$, there exist vectors $\vec{X}$ and $\vec{Y}$ such that all entries in the union of $A\vec{X}, A\vec{X} + B\vec{Y}, A\vec{X} \cdot B\vec{Y}$ are monochromatic, where $A\vec{X} \cdot B\vec{Y}$ denote the vector each of its entries are pointwise multiplication of the coordinates of $A\vec{X} \text{ and } B\vec{Y}$. In this article, we give a short combinatorial proof of this result, and then we extend it to infinite image partition regular matrices.

math.CO↗
arXiv2026

Explicit enumeration and large-valence asymptotics of even-valent maps

Let $\mathscr{N}_g(2ν,j)$ denote the number of connected labelled $2ν$-valent maps of genus $g$ with $j$ vertices. Explicit bivariate formulae for $\mathscr{N}_g(2ν,j)$ have been available only in the planar and toroidal cases. Using a structural formula of Ercolani et al (2023), we translate the problem of determining an explicit bivariate formula for $\mathscr{N}_g(2ν,j)$, $g \geq 2$, to finding finitely many counts with a fixed number of vertices. For $g=2,3$ and $4$ we determine these counts using the associated orthogonal polynomials, yielding explicit bivariate formulae for $\mathscr{N}_g(2ν,j)$ in these genera. Furthermore, the same method applies for every $g\ge5$ at the cost of additional computation. From these formulae we obtain the leading-order asymptotics of $\mathscr{N}_g(2ν,j)$ as $ν\to\infty$ for $g=2,3,4$, and we conjecture the structure of these formulae in general genus. In addition, we establish an analogous reduction to finitely many counts, derive explicit formulae and large-valence asymptotics, and formulate corresponding conjectures for two-legged even-valent maps.

math.CO↗