Almost-Orthogonality in Lp Spaces: A Case Study with Grok

Problem
This paper addresses a gap in the understanding of the triangle inequality in ( L^p ) spaces, specifically for ( p > 2 ). The authors present a counterexample to a proposed inequality by Carbery, demonstrating its failure in this regime. This work is a preprint and has not undergone peer review, indicating that the findings should be interpreted with caution.

Method
The authors construct a counterexample to Carbery’s inequality, which posits a specific form of the triangle inequality for functions in ( L^p ) spaces. They show that the inequality fails for all ( p > 2 ). Subsequently, they derive a necessary condition for the inequality to hold, establishing that the exponent ( c ) must satisfy ( c \le p’ ). At the critical exponent ( c = p’ ), they prove the inequality for all integer values ( p \ge 2 ). In the second part, they derive a sharp three-function bound that incorporates a quantification of orthogonality among the functions involved, represented by ( \Gamma ). The bound is expressed as:

[ \Big|\sum_{j=1}^{3} f_j\Big|p \le \left(1 + 2\Gamma^{c(p)}\right)^{1/p’} \Big(\sum{j=1}^{3} |f_j|_p^p\Big)^{1/p} ]

where ( c(p) = \frac{2\ln(2)}{(p-2)\ln(3) + 2\ln(2)} ) is shown to be optimal. The authors also reference the use of the large language model Grok in deriving some intermediate lemmas and inequalities.

Results
The authors demonstrate that their derived inequality holds for all integer ( p \ge 2 ) and specifically for ( p \ge 3 ) in the three-function case. They improve upon the previous exponent ( r(p) = \frac{6}{5p-4} ) established by Carlen, Frank, and Lieb, providing a more refined bound that is optimal. The paper does not provide specific numerical results or effect sizes against named baselines, as the focus is on theoretical advancements rather than empirical validation.

Limitations
The authors acknowledge that their results are limited to integer values of ( p ) and do not extend to non-integer ( p ). Additionally, the implications of their findings for practical applications in functional analysis or related fields are not fully explored. The reliance on Grok for intermediate results raises questions about the reproducibility and verification of these findings without further empirical validation.

Why it matters
This work has significant implications for the field of functional analysis, particularly in understanding the behavior of inequalities in ( L^p ) spaces. The results could influence future research on the structure of function spaces and the development of new inequalities. Furthermore, the optimality of the derived bounds may lead to advancements in applications where such inequalities are critical, such as in signal processing, data analysis, and machine learning frameworks that utilize ( L^p ) norms.

Authors: Ziang Chen, Jaume de Dios Pont, Paata Ivanisvili, Jose Madrid, Haozhu Wang
Source: arXiv:2605.05192
URL: https://arxiv.org/abs/2605.05192v1