Elliptope (${\mathcal{E}}_N=\{X⪰0:\mathrm{diag}X=1\}$)

The set of real positive semidefinite matrices with unit diagonal. Equivalently, its members are Gram matrices of unit vectors, by the real spectral theorem. Every two-by-two principal minor gives $|X_{ij}|\le 1$, so the set is bounded; it is also closed, hence compact in finite dimension. This gives attainment of linear objectives used in semidefinite programming.