OP-00145 Partially resolved

Dimension-independent Hlawka constants for Schatten norms

For $1 \le p < \infty$ let $\|\cdot\|_p$ be the Schatten $p$-norm on complex $N \times M$ matrices, and let $D_p(U, V) = \|U\|_p + \|V\|_p - \|U + V\|_p$. Let $C_p$ be the least $C \ge 0$ such that $$\|X\|_p + \|Y\|_p + \|Z\|_p - \|X + Y + Z\|_p \le C \bigl(D_p(X,Y) + D_p(X,Z) + D_p(Y,Z)\bigr)$$ for all positive integers $N, M$ and all $X, Y, Z \in \mathbb{C}^{N \times M}$, including those with $D_p(X,Y) + D_p(X,Z) + D_p(Y,Z) = 0$, and $C_p = \infty$ if there is no such $C$. For $p = 2$ and $C = 1$ the inequality is Hlawka's inequality, and $C_2 = 1$. Audenaert and Kittaneh showed that no finite constant works for the operator norm, and reported numerical simulations showing $C_1 \ge 40$.

Problem. Determine $C_p$ for every $1 \le p < \infty$.

Source: Audenaert and Kittaneh (2017), Problem 6 (Problem 7 in arXiv:1201.5232, 2012).

Imported from github.com/ajt60gaibb/OpenProblemsInNLA.

Classification

Primary: FA — Functional Analysis
Secondary: NA
Keywords: Hlawka-type inequalities, norm inequalities, OpenProblemsInNLA, Schatten norms
Literature checked: Oct. 6, 2026 by clemens-admin

References

@incollection{audenaert2017,
  author        = {Audenaert, Koenraad M. R. and Kittaneh, Fuad},
  title         = {Problems and conjectures in matrix and operator inequalities},
  booktitle     = {{\'E}tudes op{\'e}ratorielles},
  series        = {Banach Center Publ.},
  volume        = {112},
  publisher     = {Polish Acad. Sci. Inst. Math.},
  address       = {Warsaw},
  year          = {2017},
  pages         = {15--31},
  doi           = {10.4064/bc112-0-2},
  eprint        = {1201.5232},
  archivePrefix = {arXiv},
}
@misc{openproblemsinnla-mi25,
  author       = {Townsend, Alex},
  title        = {{MI-25}: Dimension-independent Hlawka constants for Schatten norms},
  howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
  year         = {2026},
  url          = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/matrix-inequalities-and-norms/MI-25/README.md},
  note         = {Read at commit e375a6f, 2026-09-30},
}

Resolution claims

The trace-norm Hlawka constant is infinite v1 GitHub ↗

Matthew J. Colbrook

For a norm, let $N$ be the three-summand defect $\sum_j\|X_j\|-\|X_1+X_2+X_3\|$ and $D$ the sum of the three pairwise defects $\|X_i\|+\|X_j\|-\|X_i+X_j\|$. For the trace norm (Schatten $p=1$), no finite constant $C_1$ gives $N\le C_1D$. The proof is an explicit family: three unit vectors $u_j\in\mathbb R^2$ at mutual angles of $120^\circ$, lifted to $v_j=(\sqrt{1-t}\,u_j,\sqrt t)\in\mathbb R^3$, give rank-one matrices $X_j=u_jv_j^{\mathsf T}$; for …

Partial resolution · not verified · fidelity not established · posted on GitHub 11 Sep 2026

Status history

When	Change	By	Reason
2026-10-08	Open → Partially resolved	The registry's curators	The trace-norm Hlawka constant is infinite v1 recorded from OpenProblemsInNLA @ eb7c9d6, MI-25/solution.pdf.

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-00145, "Dimension-independent Hlawka constants for Schatten norms", clopen.solutions.