Abstract
We study and analyze the algebraic structure of the arbitrary-order cones. We show that, unlike popularly perceived, the arbitrary-order cone is self-dual for any order greater than or equal to 1. We establish a spectral decomposition, consider the Jordan algebra associated with this cone, and prove that this algebra forms a Euclidean Jordan algebra with a certain inner product. We generalize some important notions and properties in the Euclidean Jordan algebra of the second-order cone to the Euclidean Jordan algebra of the arbitrary-order cone.
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