Cross-polytope

2008, Robert Erdahl, Andrei Ordine, Konstantin Rybnikov, Perfect Delaunay Polytopes and Perfect Quadratic Functions on Lattices, Matthias Beck, et al. (editors), Contemporary Mathematics 452: Integer Points in Polyhedra — Geometry, Number Theory, Representation Theory, Algebra, Optimization, Statistics, AMS-IMS-SIAM Joint Summer Conference, June 11-15, 2006, page 104, The convex hulls of such crosses often appear as cells in Delaunay tilings – cross polytopes are examples, as are the more spectacular symmetric perfect Delaunay polytopes.

The case p = 1, which is actually important to us, is illustrated in Figure 1.14; we are looking at the intersection of a cross-polytope with the probability simplex.

The cross polytope βₙ is the regular polytope in n dimensions corresponding to the convex hull of the points formed by permuting the coordinates (±1,0,0,...,0), and has therefore 2n vertices. It is named so because its vertices are located equidistant from the origin, along the Cartesian axes in n-space. The cross polytope in n dimensions is bounded by 2ⁿ (n - 1)-simplices, has 2n vertices and 2ⁿ(n - 1) edges.

If one works with the l₁ norm, a ball (the set of all points whose distance lies within a certain radius around a point of interest) has the shape of a cross-polytope. A one-dimensional cross-polytope is a line segment, a two-dimensional cross-polytope is a square, for three dimensions, an octahedron, and so forth.

While centrally symmetric d-polytopes with 2d vertices are affinely equivalent to the d-dimensional cross-polytope, a complete classification of the centrally symmetric d-polytopes with 2d + 2 vertices is out of reach.