Short curves in homogeneous spaces of the unitary group for unitarily invariant norms
Abstract
Let A be a unital *-subalgebra of Mn(C) and let M = U(n)/U(A), with the quotient Finsler metric induced by a unitarily invariant norm on u(n), normalized so that ||iP|| = 1 for rank-one projections P. This family contains the unitary orbits of Hermitian matrices, in particular flag manifolds and Grassmann manifolds. We prove that if z0 is a minimal lifting of a tangent vector X with ||X||∞ < π/2, then the one-parameter group t ↦ et z0· o has minimal length among all piecewise smooth curves in M joining its endpoints. The same holds in manifolds N=SU(n)/(SU(n) ∩ A) for every Ad-invariant Finsler norm on su(n), not necessarily symmetric (i.e. the unit sphere does not need to be balanced). The proof relies on the convexity of the distance from the identity along geodesic segments, valid for all these norms. We also prove that the ball of radius π/2 around the identity, for the spectral norm, is convex along one-parameter segments. This extends known results for the p-Schatten norms with p even, and gives a new proof for the operator norm, where the result is due to Duran, Mata-Lorenzo and Recht.
Related
- Clarkson--McCarthy type inequalities, part II: refinements, unitary orbits, unitarily invariant norms and ℓs--ℓs estimates
- Measure Convergence under Weak Domination for Unitarily Invariant Norms on Finite von Neumann Algebras
- Universal optimality of the double-centred matrix under unitarily invariant norms for dissimilarity data
- Unitarily invariant norms
- A Sharp Unitarily Invariant Norm Bound for the Off-Diagonal Block Perturbation of a Hermitian Matrix
BibTeX
@misc{larotonda2026short,
title = {{Short curves in homogeneous spaces of the unitary group for unitarily invariant norms}},
author = {Gabriel Larotonda},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31535},
eprint = {2610.31535},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31535},
}
RIS
TY - UNPB
TI - Short curves in homogeneous spaces of the unitary group for unitarily invariant norms
AU - Gabriel Larotonda
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31535
UR - https://arxiv.org/abs/2610.31535
AB - Let A be a unital *-subalgebra of M_n(C) and let M = U(n)/U(A), with the quotient Finsler metric induced by a unitarily invariant norm on u(n), normalized so that ||iP|| = 1 for rank-one projections P. This family contains the unitary orbits of Hermitian matrices, in particular flag manifolds and Grassmann manifolds. We prove that if z_0 is a minimal lifting of a tangent vector X with ||X||_∞ < π/2, then the one-parameter group t ↦ e^t z_0· o has minimal length among all piecewise smooth curves in M joining its endpoints. The same holds in manifolds N=SU(n)/(SU(n) ∩ A) for every Ad-invariant Finsler norm on su(n), not necessarily symmetric (i.e. the unit sphere does not need to be balanced). The proof relies on the convexity of the distance from the identity along geodesic segments, valid for all these norms. We also prove that the ball of radius π/2 around the identity, for the spectral norm, is convex along one-parameter segments. This extends known results for the p-Schatten norms with p even, and gives a new proof for the operator norm, where the result is due to Duran, Mata-Lorenzo and Recht.
ER -