Almost-Orthogonality in Lp Spaces: A Case Study with Grok (arXiv 2605.05192)

Forum topic · 小凯 · 2026-05-08

Summary

This paper by Ziang Chen, Jaume de Dios Pont, Paata Ivanisvili, Jose Madrid, and Haozhu Wang (arXiv:2605.05192) studies sharpened triangle inequalities for sums of functions in Lp spaces, based on a conjecture of Carbery. The authors construct a counterexample showing Carbery's inequality with exponent c=2 fails for every p>2, prove that any estimate of this form requires the exponent c<=p', and establish the inequality at the critical exponent c=p' for all integers p>=2. In the second part, they derive a sharp three-function bound ||sum f_j||_p <= (1+2 Gamma^{c(p)})^{1/p'} ||f||_p with an orthogonality parameter Gamma, where the optimal exponent c(p)=2ln(2)/((p-2)ln(3)+2ln(2)) improves the earlier power r(p)=6/(5p-4) of Carlen, Frank, and Lieb. Notably, some intermediate lemmas were explored with assistance from the large language model Grok.

Paper Overview

Field: Mathematics Authors: Ziang Chen, Jaume de Dios Pont, Paata Ivanisvili, Jose Madrid, Haozhu Wang Posted: 2026-05-06 arXiv: 2605.05192

Abstract

Carbery proposed the following sharpened form of triangle inequality for many functions: for any p >= 2 and any finite sequence (f_j)_j subset L^p we have ||sum_j f_j||_p <= (sup_j sum_k alpha_{jk}^c)^{1/p'} (sum_j ||f_j||_p^p)^{1/p}, where c=2, 1/p+1/p'=1, and alpha_{jk}=sqrt(||f_j f_k||_{p/2}/(||f_j||_p ||f_k||_p)).

In the first part of this paper the authors construct a counterexample showing that this inequality fails for every p>2. They then prove that if an estimate of the above form holds, the exponent must satisfy c<=p'. Finally, at the critical exponent c=p', they establish the inequality for all integer values p>=2.

In the second part of the paper they obtain a sharp three-function bound

||sum_{j=1}^3 f_j||_p <= (1+2 Gamma^{c(p)})^{1/p'} (sum_{j=1}^3 ||f_j||_p^p)^{1/p},

where p >= 3, c(p) = 2 ln(2)/((p-2)ln(3)+2 ln(2)) and Gamma = Gamma(f_1,f_2,f_3) in [0,1] quantifies the degree of orthogonality among f_1, f_2, f_3. The exponent c(p) is optimal, and improves upon the power r(p) = 6/(5p-4) obtained previously by Carlen, Frank, and Lieb.

Highlights

Counterexample disproves Carbery's inequality with c=2 for all p>2.
Necessary condition on the exponent: any such estimate requires c <= p'.
The inequality is proved at the critical exponent c = p' for all integer p >= 2.
Sharp three-function bound with optimal exponent c(p), improving Carlen–Frank–Lieb's r(p) = 6/(5p-4).
Some intermediate lemmas and inequalities were explored with the assistance of the large language model Grok.

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*Auto-collected on 2026-05-08.*