Hlawka constants for Schatten -norms (Audenaert–Kittaneh Problem 7)
Hlawka's inequality says that in an inner product space the "triple deficit" is at most the sum of the three "pair deficits". Audenaert and Kittaneh asked for the best constant , independent of the matrix size, if it exists, such that for all complex matrices
and in particular for the best constant when the norm is the Schatten -norm. They noted that by Hlawka's inequality and that is infinite. They reported that numerical simulations give , "indicating that it might be arbitrarily large as well".
- Result
- Proved(see note)
- Status
- Partial result
- AI contribution
- AI co-developed
- Method
- Argument
- Field
- Matrix and operator inequalities; Schatten norms
- Posed by
- Koenraad Audenaert and Fuad Kittaneh, Problem 7 in "Problems and Conjectures in Matrix and Operator Inequalities" (arXiv:1201.5232)
- Year posed
- 2012
- Years open
- 14y
- Solved
- 2026-09-22
- Model
- GPT 5.6 Sol and GPT 6 Astra
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Lean-verified
- Publication
- Announced
- Significance
- 20 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Settles the "if it exists" part for every : a finite size-independent constant exists exactly when . The proof gives some constant, not the best one, for complex matrices of every size, square or rectangular.
At no constant exists, already for matrices, turning Audenaert and Kittaneh's numerical into . At none exists on matrices, as they already knew.
For diagonal matrices and the best constant is an explicit , the maximum of a ratio over the family , , with . Since diagonal matrices are matrices, for .
Open: the best for general matrices (except ) and the best diagonal constant for ; the cutoff 256 is not claimed optimal. All of this is in Lean except the one-line step from diagonal to general matrices.
What the AI did
The author and AI agents co-developed both results. Existence: GPT 5.6 Sol, running in Codex, built the Lean proof in small kernel-checked layers from audited mathematical proof artifacts; the approach was motivated by the Ball–Carlen–Lieb uniform convexity framework. Sharp diagonal constant: the argument came out of two weeks of directed proof search. The author identified the proof strategy, diagnosed repeated dead ends where agents stalled, and refined the approach through failure analysis at each stage. AI agents executed computational searches and produced candidate lemmas, and GPT 6 Astra, running in Codex, formalized the final argument in Lean. The author, GPT 5.6 Sol and Claude Fable 5.1 reviewed the work. The proofs have not been examined in depth by human experts.
Verification
Audited here on 27 September 2026 from a clone at commit 1829591: 48 Lean files, 8,711 lines on Lean 4.35.0-rc2; exactly four sorry, all four the deliberate holes in the two Challenge files that the solution files fill, which is the comparator layout Palomar expects; zero axiom declarations, zero native_decide. CI run 35949577091 is green there and audits the four headline theorems to propext, Classical.choice and Quot.sound.
The submitter notes Palomar last passed its statement check at the earlier commit 79aa498, before the toolchain upgrade, which would normally leave the statements unanchored here. So both Challenge files were fetched at both commits and compared. All four theorem statements are byte-identical. Of the supporting definitions three differ, all cosmetically: N renamed size, and pairGapSum refactored through a new pairGap whose body is size x + size y - size (x + y), the same sum of three pair deficits. The anchoring carries over, which is what lean-verified requires.
Not covered by the four statements, as the submitter says: the bounds 939p/2000 < K_p <= p, library theorems, and the diagonal-to-general step, not in Lean. No human expert has examined the proofs.
Sources
Submitted by ZestyWombat854 on