Internal approximations of reachable sets of control systems with state constraints

We construct self-intersecting flexible cross-polytopes in the spaces of constant curvature, that is, in Euclidean spaces (mathbb{E}^n) , spheres (mathbb{S}^n) , and Lobachevsky spaces Λ ⁿ of all dimensions n. In dimensions n ≥ 5, these are the first examples of flexible polyhedra. Moreover, we classify all flexible cross-polytopes in each of the spaces (mathbb{E}^n) , (mathbb{S}^n) , and Λ ⁿ . For each type of flexible cross-polytopes, we provide an explicit parametrization of the flexion by either rational or elliptic functions.