Equipartite polytopes (Q607856)

The authors call a polytope equipartite if, for any partition of its vertex set into two equal-size sets \(V\) and \(W\), there is an isometry of the polytope that maps \(V\) onto \(W\). For every \(d\geq 2\), the main results read as follows: (1) an equipartite \(d\)-polytope has at most \(2d+2\) vertices and (2) there is an equipartite \(d\)-polytope with \(2d+2\) vertices. Besides, the authors construct several examples of equipartite \(d\)-polytopes and conjecture that the list is complete.