Four-Dimensional Semi-Homogeneous Convex Cones: A Complete Classification
Abstract
We classify all four-dimensional semi-homogeneous convex cones, i.e., closed, pointed, full-dimensional cones whose connected automorphism group has principal orbits of codimension at most one. On such cones, the Monge-Ampère equation defining the canonical barrier reduces to an ordinary differential equation. This makes this barrier computationally accessible. The semi-homogeneous cones can then be used in modeling by conic programs. An obvious subset of such cones are the homogeneous cones. The key point allowing the classification is that for all other cones, the automorphism group has a three-dimensional subgroup acting locally faithfully on the principal orbits. This subgroup must have a Lie algebra which is either abelian or a direct product of a 1-dimensional algebra and a two-dimensional algebra generated by A,B satisfying [A,B]=B. In both cases we classify all 4-dimensional representations up to isomorphism, and for each representation all invariant convex cones. For every cone an orbit invariant is computed.
Related
- One-Dimensional Symmetry Reductions of the Monge-Ampère Equation on Some Semi-Homogeneous Convex Cones in ℝ4
- A complete parameter classification of left-invariant Bott affine Ricci solitons for two rotated distributions on three-dimensional Lorentzian Lie groups
- A complete classification of the existence of finite-dimensional quantum solutions to inconsistent linear constraint systems
- Sharp Weighted Sobolev Trace Inequalities on Convex Cones and Rigidity of Extremals
- Monotone invariant valuations on convex cones
BibTeX
@misc{valiakhmetov2026fourdimensional,
title = {{Four-Dimensional Semi-Homogeneous Convex Cones: A Complete Classification}},
author = {Amir Valiakhmetov and Ewsey Obzherin and Alexey Chernov and Roland Hildebrand},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00101},
eprint = {2611.00101},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00101},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Four-Dimensional Semi-Homogeneous Convex Cones: A Complete Classification
AU - Amir Valiakhmetov
AU - Ewsey Obzherin
AU - Alexey Chernov
AU - Roland Hildebrand
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00101
UR - https://arxiv.org/abs/2611.00101
AB - We classify all four-dimensional semi-homogeneous convex cones, i.e., closed, pointed, full-dimensional cones whose connected automorphism group has principal orbits of codimension at most one. On such cones, the Monge-Ampère equation defining the canonical barrier reduces to an ordinary differential equation. This makes this barrier computationally accessible. The semi-homogeneous cones can then be used in modeling by conic programs. An obvious subset of such cones are the homogeneous cones. The key point allowing the classification is that for all other cones, the automorphism group has a three-dimensional subgroup acting locally faithfully on the principal orbits. This subgroup must have a Lie algebra which is either abelian or a direct product of a 1-dimensional algebra and a two-dimensional algebra generated by A,B satisfying [A,B]=B. In both cases we classify all 4-dimensional representations up to isomorphism, and for each representation all invariant convex cones. For every cone an orbit invariant is computed.
ER -