Carbery’s proposed Lp exponent bound
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What exponent can strengthen the Lp triangle inequality in Carbery’s almost-orthogonality question? For p>2, the proposed value is c=2; p′=p/(p−1) is the competing necessary upper constraint on c.
Why it matters and what counts as progress
Why it matters: The exponent controls a proposed strengthening of almost-orthogonality; identifying its necessary limit distinguishes a structural obstruction from a numerical example.
What counts as progress: Give a structural counterexample to a proposed exponent or prove an inequality attaining the optimal admissible exponent.
How the problem developed
Anthony Carbery Formulated almost-orthogonality estimates in Schatten and Lp settings, including conjectured exponents for sums of functions with disjoint Fourier support.
- 2020 · Partial result [2]
Carlen, Frank, and Lieb Established a three-function almost-orthogonality estimate with exponent r(p)=6/(5p−4), a bound later improved in the 2026 work.
- 6 May 2026 · Counterexample and bound [2]
Chen, de Dios Pont, Ivanisvili, Madrid, and Wang, with Grok Reported that the proposed exponent 2 fails for every p>2, proved p′ is necessary, established the p′ inequality for integer p≥2, and gave an optimal three-function exponent for p≥3.
AI contribution
Almost-Orthogonality in Lp Spaces: A Case Study with Grok
For p>2, a common-set/private-set construction shows Carbery’s proposed exponent c=2 fails. Grok also supplied the kernel inequality used to establish the optimal necessary constraint c≤p′.
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