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  "body": "\n## Definition\n\nLet $P$ be a [[convex polytope|300K6CB1KH]] of dimension $d \\ge 1$.\nAn edge of $P$ is a face of dimension $1$.\n$P$ is *simple* if every [[vertex|R7V3S9V891]] of $P$ lies in exactly $d$ edges of $P$.\n\n## Remarks\n\nEquivalently, every vertex of $P$ lies in exactly $d$ [[facets|DPDXN2FJC4]] of $P$.\nEvery vertex of a $d$-polytope lies in at least $d$ edges, so simple polytopes are those where this lower bound is always reached.\n\nExample: the cube $[0,1]^3$ is simple.\nAt 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.\n\nNon-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 $\\pm e_3$.\n\nIf $P$ contains the origin in its interior, then $P$ is simple if and only if its polar polytope is simplicial.\nEvery polygon is simple.\n",
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