PreprintCC BY 4.0arXiv
One-Dimensional Symmetry Reductions of the Monge-Ampère Equation on Some Semi-Homogeneous Convex Cones in ℝ4
Abstract
Motivated by non-symmetric conic optimization, we construct six explicit families of semi-homogeneous proper convex cones K in ℝ4. For each family, a specified three-dimensional subgroup of linear automorphisms of K has generic orbits of codimension one. This yields an orbit invariant t and reduces the Monge-Ampère equation det F''=e2F, defining the canonical barrier on K, to an ordinary differential equation (ODE) with independent variable t. The families include asymmetric power, rotational-power, exponential, and nonabelian-type cones. For the power and exponential cases we derive first integrals and parametric solutions of the ODE. The reductions open the way for computing the canonical barrier on these cones numerically.
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BibTeX
@misc{valiakhmetov2026onedimensional,
title = {{One-Dimensional Symmetry Reductions of the Monge-Ampère Equation on Some Semi-Homogeneous Convex Cones in $\mathbbR^4$}},
author = {Amir Valiakhmetov and Ewsey Obzherin and Alexey Chernov and Roland Hildebrand},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26405},
eprint = {2610.26405},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26405},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - One-Dimensional Symmetry Reductions of the Monge-Ampère Equation on Some Semi-Homogeneous Convex Cones in ℝ^4
AU - Amir Valiakhmetov
AU - Ewsey Obzherin
AU - Alexey Chernov
AU - Roland Hildebrand
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26405
UR - https://arxiv.org/abs/2610.26405
AB - Motivated by non-symmetric conic optimization, we construct six explicit families of semi-homogeneous proper convex cones K in ℝ^4. For each family, a specified three-dimensional subgroup of linear automorphisms of K has generic orbits of codimension one. This yields an orbit invariant t and reduces the Monge-Ampère equation det F''=e^2F, defining the canonical barrier on K, to an ordinary differential equation (ODE) with independent variable t. The families include asymmetric power, rotational-power, exponential, and nonabelian-type cones. For the power and exponential cases we derive first integrals and parametric solutions of the ODE. The reductions open the way for computing the canonical barrier on these cones numerically.
ER -