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I know that a (d dimensional) simple polytope is defined as one such that each vertex is contained in exactly d facets.

I heard that an equivalent characterization is that the set of outward edges from any given vertex form a basis for the ambient space.

I’m not sure how to prove this. I know that there will be d edges containing this vertex: since the polytope is simple, each vertex is the intersection of d facets, and so each choice of d-1 of these facets will form a unique edge, and there are d choices of d-1 facets.

However I’m not sure how to show these edges are in linearly independent directions for starters.

Nor am I sure how to prove that if a polytope has this property: that each vertex is contained in edges whose direction vectors form a basis for the space, should imply that this vertex is contained in exactly d facets.

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