The maximal volume of projections of the cross-polytope

What the math gives to ML

The paper proves a sharp geometric inequality: the volume of the absolute convex hull generated by vectors is bounded by the determinant of their frame operator, with equality after normalization only for an orthonormal spanning family. This suggests a global diversity regularizer for learned prototypes, expert vectors, or embeddings that penalizes anisotropic or collapsed representations through a log-determinant objective. Unlike pairwise repulsion, the determinant directly measures whether the vectors collectively span all representation directions. The first implementation should compare this frame-isotropy penalty with pairwise orthogonality regularization at equal computational cost.