Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2608.26395

Isoperimetric problems related to extremal diameter graphs in 3D: theoretical and numerical aspects

Beniamin Bogosel
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study optimization problems for separable functionals of the Euclidean or spherical lengths of dual edge pairs in finite extremal unit-diameter configurations in three dimensions. For a fixed diameter graph, these problems lead to nonconvex constrained optimization of the vertex coordinates. We first prove that a convergent sequence of extremal configurations retains an extremal geometric core after coincident points are merged and vertices incident to at most one diameter are removed. A spherical Crofton argument then gives a sharp lower bound for additive concave functionals, attained by the regular tetrahedron. We also analyze the effect of inserting or deleting dangling vertices. For the sum of products of spherical dual-edge lengths, we obtain an exact supremal reformulation of the three-dimensional Blaschke--Lebesgue area problem. The numerical study uses all 10,644 available extremal configurations with at most 16 vertices. We combine direct evaluation with gradient-based local optimization on each fixed graph and reconstruct the intrinsic diameter graph and its dual pairs after vertex collisions. For every supplied graph, the selected verified endpoint contains a regular tetrahedron. These computations do not certify the global maximum for any fixed graph, but they motivate structural conjectures connecting tetrahedral containment with the Blaschke--Lebesgue problem.

Explore related subjects

math.OCmath.MG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Beniamin Bogosel. 2026-08-26. Isoperimetric problems related to extremal diameter graphs in 3D: theoretical and numerical aspects. https://arxiv.org/abs/2608.26395

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

KEEP EXPLORING

Related papers

arXiv2026

Genericity of Polyak-Lojasiewicz Inequalities for Entropic Mean-Field Neural ODEs

We address the behavior of idealized deep residual neural networks (ResNets), modeled via an optimal control problem set over continuity (or adjoint transport) equations. The continuity equations describe the statistical evolution of the features in the asymptotic regime where the layers of the network form a continuum. The velocity field is expressed through the network activation function, which is itself viewed as a function of the statistical distribution of the network parameters (weights and biases). From a mathematical standpoint, the control is interpreted in a relaxed sense, taking values in the space of probability measures over the set of parameters. We investigate the optimal behavior of the network when the cost functional arises from a regression problem and includes an additional entropic regularization term on the distribution of the parameters. In this framework, we focus in particular on the existence of stable optimizers --that is, optimizers at which the Hessian of the cost is non-degenerate. We show that, for an open and dense set of initial data, understood here as probability distributions over features and associated labels, there exists a unique stable global minimizer of the control problem. Moreover, we show that such minimizers satisfy a local Polyak--Lojasiewicz inequality, which can lead to exponential convergence of the corresponding gradient descent when the initialization lies sufficiently close to the optimal parameters. This result thus demonstrates the genericity (with respect to the distribution of features and labels) of the Polyak--Lojasiewicz condition in ResNets with a continuum of layers and under entropic penalization.

math.OC↗
arXiv2026

A regret minimization approach to fixed-point iterations

We propose a conversion scheme that turns regret minimizing algorithms into fixed point iterations, with convergence guarantees following from regret bounds. The resulting iterations can be seen as a grand extension of the classical Krasnoselskii--Mann iterations, as the latter are recovered by converting the Online Gradient Descent algorithm. This approach yields new simple iterations for finding fixed points of non-self operators. We also focus on converting algorithms from the AdaGrad family of regret minimizers, and thus obtain fixed point iterations with adaptive guarantees of a new kind. Numerical experiments on various problems demonstrate faster convergence of AdaGrad-based fixed point iterations over Krasnoselskii--Mann iterations.

math.OC↗
arXiv2026

Variational Analysis in Spectral Decomposition Systems

This work is concerned with the variational analysis of functions defined on Euclidean spaces whose values depend solely on certain invariants (``spectrum'') of their arguments, a class we term ``spectral functions.'' Building on our previous work \cite{PartI} on the convex analysis of such functions, we work in the abstract framework of spectral decomposition systems, which covers a wide range of previously studied settings, including eigenvalue decomposition of Hermitian matrices and singular value decomposition of rectangular matrices, and allows the derivation of new results in more general settings such as normal decomposition systems. The main results of this work provide constructive formulae for computing the regular, limiting, and Clarke subdifferentials of a spectral function in terms of the corresponding objects of the associated invariant function. Finally, we obtain a generalization of Lidski\uı's theorem on the spectrum of additive perturbations of Hermitian matrices to arbitrary spectral decomposition systems.

math.OC↗