crosspolytopes
The cross polytope, also called the orthoplex, is a regular convex polytope in n dimensions. It is the dual of the n-cube and can be described as the convex hull of the 2n points ±e1, ±e2, ..., ±en in R^n, where ei are the standard basis vectors. Equivalently, it is the unit ball of the L1 norm, given by the inequality sum_i |x_i| ≤ 1.
- Vertices: 2n, consisting of the pairs {ei, −ei} for i = 1,...,n.
- Facets: 2^n, each an (n−1)-simplex.
- Edges: connect each vertex to all vertices except its opposite; the 1-skeleton is the complete multipartite
- Symmetry: the symmetry group is the hyperoctahedral group, of order 2^n n!, arising from permutations of
- Dual relationship: the cross polytope is the polar dual of the n-cube, and vice versa.
- In the Schläfli notation, the regular n-dimensional cross polytope has the symbol {3,3,...,4} (with n−1 entries
- In low dimensions, it appears as the square in 2D (a diamond), the regular octahedron in 3D,
- The unit cross polytope defined by sum_i |x_i| ≤ 1 has volume 2^n / n!.
- The vertices lie on a common inscribed sphere, reflecting its regularity, while the dual relationship to
- The cross polytope serves as a geometric model for L1 optimization, sparse approximation, and various areas