Sonotopes

Sonotopes are a class of mathematical objects that generalize the concept of polytopes to higher dimensions. While polytopes are defined in Euclidean space, sonotopes are defined in a more general setting known as a Jordan algebra. This generalization allows for the study of sonotopes in a variety of contexts, including non-Euclidean geometries and complex numbers.

Sonotopes were first introduced by Jean Dieudonné in the 1950s as a way to study the geometry of Jordan algebras. A sonotope is defined as the convex hull of a finite set of points in a Jordan algebra. The study of sonotopes has since become an active area of research in mathematics, with applications in areas such as algebraic geometry, representation theory, and optimization.

One of the key properties of sonotopes is that they are self-dual, meaning that their polar dual is isomorphic to the original sonotope. This property makes sonotopes useful in the study of duality in geometry. Additionally, sonotopes have been shown to have interesting combinatorial properties, such as the fact that their face lattice is a distributive lattice.

Sonotopes have also been studied in the context of optimization problems, where they are used to model convex sets in Jordan algebras. This has led to the development of new algorithms for solving optimization problems in non-Euclidean geometries.

In summary, sonotopes are a generalization of polytopes to Jordan algebras, with applications in algebraic geometry, representation theory, and optimization. They have interesting geometric and combinatorial properties, and their study continues to be an active area of research in mathematics.