Spectral Properties of Convex Bodies

We use the Madden-Robertson classification of regular convex bodies to show that convex bodies are spectral and strongly symmetric if and only if they are affinely isomorphic to the normalized state spaces of simple euclidean Jordan algebras, or to simplices. Further, we discuss the relevance of this result for general probabilistic theories of quantum and classical physical systems, and its relation to other characterizations of various classes of euclidean Jordan algebra state spaces.