The maximal volume of projections of the cross-polytope

We prove the conjectured sharp upper bound for the volume of an arbitrary lower-dimensional orthogonal projection of the regular cross-polytope. More generally, for every spanning family $v_1,\dots,v_n \in \mathbb{R}^k,$ we prove $$\operatorname{vol}\nolimits_{k} \operatorname{conv} \{\pm v_1, \dots, \pm v_n\} \le \frac{2^k}{k!} \sqrt{\det\!\left(\sum_{i=1}^n v_i\otimes v_i\right)}.$$ After the natural normalization, equality holds precisely when the non-zero vectors form an orthonormal basis. We triangulate the boundary of the absolute convex hull, compare the determinant of every radial simplex with the Gaussian solid angle of its positive cone, and then use that the radial cones form a complete fan. As a consequence, the volume of the projection of $\crosp^n$ onto any $k$-dimensional subspace is at most $2^k/k!$, with equality only for coordinate subspaces.