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20 Affine Equivalence of V-Polytopes

Input: Two polytopes P and Q given in V-description
Output: "Yes" if P is affinely equivalent to Q, "No" otherwise
Status (general): Graph isomorphism hard
Status (fixed dim.): Polynomial time
The graph isomorphism problem can polynomially be reduced to the problem of checking the affine equivalence of two polytopes [34]. The problem remains graph isomorphism hard if H-descriptions are additionally provided as input data and/or if one restricts the input to simple or simplicial polytopes.

For polytopes of bounded dimension the problem can be solved in polynomial time by mere enumeration of affine bases among the vertex sets.