{"@context":"https://schema.org","@type":"CreativeWork","@id":"https://forgecascade.org/public/capsules/24e8177d-5bb3-4de9-b171-5ff62051178b","name":"Almost-Orthogonality in Lp Spaces: A Case Study with Grok","text":"# Almost-Orthogonality in Lp Spaces: A Case Study with Grok\n\n**Authors:** Ziang Chen, Jaume de Dios Pont, Paata Ivanisvili, Jose Madrid, Haozhu Wang\n**arXiv:** https://arxiv.org/abs/2605.05192v1\n**Published:** 2026-05-06T17:54:51Z\n\n## Abstract\nCarbery proposed the following sharpened form of triangle inequality for many functions: for any $p\\ge 2$ and any finite sequence $(f_j)_j\\subset L^p$ we have \\[ \\Big\\|\\sum_j f_j\\Big\\|_p \\ \\le\\ \\left(\\sup_{j} \\sum_{k} α_{jk}^{\\,c}\\right)^{1/p'} \\Big(\\sum_j \\|f_j\\|_p^p\\Big)^{1/p}, \\] where $c=2$, $1/p+1/p'=1$, and $α_{jk}=\\sqrt{\\frac{\\|f_{j}f_{k}\\|_{p/2}}{\\|f_{j}\\|_{p}\\|f_{k}\\|_{p}}}$. In the first part of this paper we construct a counterexample showing that this inequality fails for every $p>2$. We then prove that if an estimate of the above form holds, the exponent must satisfy $c\\le p'$. Finally, at the critical exponent $c=p'$, we establish the inequality for all integer values $p\\ge 2$.   In the second part of the paper we obtain a sharp three-function bound \\[ \\Big\\|\\sum_{j=1}^{3} f_j\\Big\\|_p \\ \\le\\ \\left(1+2Γ^{c(p)}\\right)^{1/p'} \\Big(\\sum_{j=1}^{3} \\|f_j\\|_p^p\\Big)^{1/p}, \\] where $p \\geq 3$, $c(p) = \\frac{2\\ln(2)}{(p-2)\\ln(3)+2\\ln(2)}$ and $Γ=Γ(f_1,f_2,f_3)\\in[0,1]$ quantifies the degree of orthogonality among $f_1,f_2,f_3$. The exponent $c(p)$ is optimal, and improves upon the power $r(p) = \\frac{6}{5p-4}$ obtained previously by Carlen, Frank, and Lieb. Some intermediate lemmas and inequalities appearing in this work were explored with the assistance of the large language model Grok.","keywords":["math.CA","cs.AI","math.CO","math.PR"],"about":[],"citation":[],"isPartOf":{"@type":"Dataset","name":"Forge Cascade Knowledge Graph","url":"https://forgecascade.org"},"publisher":{"@type":"Organization","name":"Forge Cascade","url":"https://forgecascade.org"}}