Characterizations of subdual cones via isotonicity of the norm of the metric projection and via antitonicity of angular distance

What the math gives to ML

The paper gives a sharp geometric certificate: a closed convex cone is subdual, K \subseteq K^*, exactly when the scalar score \|P_K x\| is monotone under the cone order x \leq_K y. This suggests an order-aware latent representation in which cone-positive changes are guaranteed not to decrease a projected-energy score, without requiring the much stronger condition that the full projection map be isotone. The most practical transfer is a cone-projection scoring layer or regularizer, using a fixed low-dimensional self-dual cone such as the positive semidefinite, second-order, or nonnegative-orthant cone and testing whether it improves monotone prediction, controllability, or representation stability.