Cross-polytope

Example

For $d \ge 1$, let $e_1, \dots, e_d$ be the standard basis of $\R^d$. The $d$-dimensional cross-polytope is $$X_d = \operatorname{conv}\set{\pm e_1, \dots, \pm e_d} = \set{x \in \R^d : \abs{x_1} + \cdots + \abs{x_d} \le 1}.$$ It is a convex polytope with the $2d$ vertices $\pm e_i$.

A set of vertices spans a proper face exactly when it contains no pair $e_i, -e_i$. Indeed, for signs $\epsilon_i \in \set{1, -1}$ the vector $c = (\epsilon_1, \dots, \epsilon_d)$ gives $c \cdot x \le 1$ on $X_d$, with equality on the hull of $\epsilon_1 e_1, \dots, \epsilon_d e_d$. So the facets are the $2^d$ simplices $\operatorname{conv}\set{\epsilon_1 e_1, \dots, \epsilon_d e_d}$, and $X_d$ is a simplicial polytope. Choosing $k + 1$ of the $d$ axes and a sign for each gives the number of $k$-faces: $$f_k(X_d) = 2^{k+1} \binom{d}{k+1}, \qquad 0 \le k \le d-1.$$