Simple polytope

Remarks

Equivalently, every vertex of $P$ lies in exactly $d$ facets of $P$. Every vertex of a $d$-polytope lies in at least $d$ edges, so simple polytopes are those where this lower bound is always reached.

Example: the cube $[0,1]^3$ is simple. At the vertex $0$ the edges are the three segments from $0$ to $e_1$, $e_2$ and $e_3$, and by symmetry every vertex lies in $3$ edges.

Non-example: the octahedron $\operatorname{conv}\set{\pm e_1, \pm e_2, \pm e_3}$ is not simple, since the vertex $e_1$ lies in the $4$ edges joining it to $\pm e_2$ and $\pm e_3$.

If $P$ contains the origin in its interior, then $P$ is simple if and only if its polar polytope is simplicial. Every polygon is simple.