Spectrahedron

In convex geometry, a spectrahedron is a shape that can be represented as a linear matrix inequality.

Alternatively, the set of n × n positive semidefinite matrices forms a convex cone in Rn × n, and a spectrahedron is a shape that can be formed by intersecting this cone with an affine subspace.

Spectrahedra are the feasible regions of semidefinite programs.

Every spectrahedral shadow is a convex set that is also semialgebraic, but the converse (conjectured to be true until 2017) is false.

[3] The spectraplex is a compact set, and can be thought of as the "semidefinite" analog of the simplex.