Simplicial polytope

Remarks

Equivalently, every proper face of $P$ is a simplex, since each face of a simplex is a simplex. Equivalently again, every facet of $P$ has exactly $d$ vertices.

Example: the octahedron $\operatorname{conv}\set{\pm e_1, \pm e_2, \pm e_3}$ in $\R^3$ is simplicial. Its facets are the eight triangles $\operatorname{conv}\set{\epsilon_1 e_1, \epsilon_2 e_2, \epsilon_3 e_3}$ with $\epsilon_i \in \set{1, -1}$, and each is the hull of $3$ affinely independent points.

Non-example: the cube $[0,1]^3$ is not simplicial, because its facets are squares with $4 > 3$ vertices.

Every polygon is simplicial, since its facets are segments. The convex hull of finitely many points in general position in $\R^d$, with at least $d+1$ of them, is simplicial.