Symmetric cone

Summary

A symmetric cone is an open convex cone in a finite-dimensional real inner product space with two properties. It equals its own dual cone, and its linear symmetries act transitively on it. The positive orthant, the Lorentz cone and the cone of positive definite symmetric matrices are the basic examples. By the Koecher–Vinberg theorem, symmetric cones are exactly the cones of squares of Euclidean Jordan algebras. They are the natural domains of semidefinite and second-order cone programming and of harmonic analysis on cones.