Year: 2026
Type: preprint
Abstract: We study two means on the five-dimensional Vinberg cone $Ω$, viewed as a sparse cone of positive definite $3\times3$ matrices. The Cholesky-Vinberg mean is obtained from the simply transitive solvable group $H$ associated with $Ω$. It is $H$-equivariant, and its interpolation curves are geodesics for two flat metric connections with torsion. One of the corresponding metrics is the canonical Hessian metric of the cone. The Log-Vinberg mean is defined by affine interpolation in the global logarithmic coordinates of the associated Vinberg algebra, or clan. The resulting Riemannian metric is complete and flat, and its weighted Fréchet means have explicit formulas. Both constructions preserve the zero pattern defining $Ω$.
Source: arXiv (Cornell University)
Author: Koufany, Khalid
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Agence Nationale de la Recherche
Awards: G1166465053, G1190173965, G6554659165
OpenAlex ID: W7212034808