One-Dimensional Symmetry Reductions of the Monge-Ampère Equation on Some Semi-Homogeneous Convex Cones in $\mathbb{R}^4$
Year: 2026
Type: preprint
Abstract: Motivated by non-symmetric conic optimization, we construct six explicit families of semi-homogeneous proper convex cones $K$ in $\mathbb R^4$. 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.
Source: arXiv (Cornell University)
Institution: Moscow Institute of Physics and Technology
Language: English
Subfield: Numerical Analysis
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223036898