Intrinsic Hilbert metrics on cones of equivalent norms

Abstract

Let X be a Banach space and let 𝒩(X) denote the family of equivalent norms on X. We study Hilbert projective metrics on its projectivization 𝒩'(X) induced by ambient cones of nonnegative functions, with particular attention to the intrinsic Hilbert metric induced by the cone 𝒩(X)∪{0}. First, we show that the symmetric logarithmic metric on 𝒩'(X) is the Hilbert projective metric induced by the cone of nonnegative real-valued functions, and we characterize the ambient cones that induce the same metric. We then study the intrinsic order on 𝒩(X). For this purpose, we introduce the triangular defect Δ_p(x,y)=p(x)+p(y)−p(x+y) and prove that two norms p,q∈𝒩(X) belong to the same intrinsic part if and only if their triangular defects are uniformly comparable, that is, aΔ_p≤Δ_q≤ bΔ_p for some a,b>0. This yields an explicit formula for the intrinsic Hilbert metric in terms of the pointwise comparison of the norms and of their triangular defects. Finally, the map Φ(p)=(p,Δ_p) realizes the intrinsic cone order of 𝒩(X) inside a canonical product cone of nonnegative functions and preserves the Hilbert metric on each intrinsic part.

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BibTeX RIS

BibTeX

@misc{acostaportilla2026intrinsic,
  title = {{Intrinsic Hilbert metrics on cones of equivalent norms}},
  author = {Juan Rafael Acosta-Portilla},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.28922},
  eprint = {2609.28922},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.28922},
}

RIS

TY  - UNPB
TI  - Intrinsic Hilbert metrics on cones of equivalent norms
AU  - Juan Rafael Acosta-Portilla
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2609.28922
UR  - https://arxiv.org/abs/2609.28922
AB  - Let X be a Banach space and let 𝒩(X) denote the family of equivalent norms on X. We study Hilbert projective metrics on its projectivization 𝒩'(X) induced by ambient cones of nonnegative functions, with particular attention to the intrinsic Hilbert metric induced by the cone 𝒩(X)∪{0}. First, we show that the symmetric logarithmic metric on 𝒩'(X) is the Hilbert projective metric induced by the cone of nonnegative real-valued functions, and we characterize the ambient cones that induce the same metric. We then study the intrinsic order on 𝒩(X). For this purpose, we introduce the triangular defect Δ_p(x,y)=p(x)+p(y)−p(x+y) and prove that two norms p,q∈𝒩(X) belong to the same intrinsic part if and only if their triangular defects are uniformly comparable, that is, aΔ_p≤Δ_q≤ bΔ_p for some a,b>0. This yields an explicit formula for the intrinsic Hilbert metric in terms of the pointwise comparison of the norms and of their triangular defects. Finally, the map Φ(p)=(p,Δ_p) realizes the intrinsic cone order of 𝒩(X) inside a canonical product cone of nonnegative functions and preserves the Hilbert metric on each intrinsic part.
ER  -

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