Characterizations of subdual cones via isotonicity of the norm of the metric projection and via antitonicity of angular distance
Abstract
Let H be a real Hilbert space, let K⊆ H be a closed convex cone, let K* be its dual closed convex cone, and let PK denote the metric projection onto K. We study scalar quantities associated with th projection PK and their interaction with the preorder induced by K. Our main result characterizes subduality by the monotonicity of the norm of a projection: K⊆ K* if and only if x≤K y implies |PKx|≤ |PKy|. We also discuss the sublinearity of the norm of a projection and of the distance from a closed convex cone, relate these functions to asymmetric seminorms, and introduce a normalized angular distance from a closed convex cone. The angular distance admits equivalent formulas in terms of PK and the ordinary distance function and, for closed convex cones with K≠ H, yields a second characterization of subduality. Finally, coordinatewise monotone norms on finite-dimensional spaces are used to aggregate distances from several closed convex cones and thereby construct asymmetric seminorms. The proofs are consequences of Moreau's decomposition theorem and the geometry of metric projections.
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BibTeX
@misc{nemeth2026characterizations,
title = {{Characterizations of subdual cones via isotonicity of the norm of the metric projection and via antitonicity of angular distance}},
author = {S. Z. Németh},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.21005},
eprint = {2608.21005},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.21005},
}
RIS
TY - UNPB
TI - Characterizations of subdual cones via isotonicity of the norm of the metric projection and via antitonicity of angular distance
AU - S. Z. Németh
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2608.21005
UR - https://arxiv.org/abs/2608.21005
AB - Let H be a real Hilbert space, let K⊆ H be a closed convex cone, let K^* be its dual closed convex cone, and let P_K denote the metric projection onto K. We study scalar quantities associated with th projection P_K and their interaction with the preorder induced by K. Our main result characterizes subduality by the monotonicity of the norm of a projection: K⊆ K^* if and only if x≤_K y implies |P_Kx|≤ |P_Ky|. We also discuss the sublinearity of the norm of a projection and of the distance from a closed convex cone, relate these functions to asymmetric seminorms, and introduce a normalized angular distance from a closed convex cone. The angular distance admits equivalent formulas in terms of P_K and the ordinary distance function and, for closed convex cones with K≠ H, yields a second characterization of subduality. Finally, coordinatewise monotone norms on finite-dimensional spaces are used to aggregate distances from several closed convex cones and thereby construct asymmetric seminorms. The proofs are consequences of Moreau's decomposition theorem and the geometry of metric projections.
ER -