Carbery’s almost-orthogonality inequality in $L^p$

Carbery posed the underlying sharpened triangle-inequality question in 2006. The many-function problem asks whether pairwise overlaps may carry power $2$ and, if not, what the largest valid power is.

Known results

• Carbery proved the corresponding two-function inequality for characteristic functions (2006).
• Carlen, Frank, Ivanisvili, and Lieb established the general two-function refinement for real $p$ (2020).
• For any finite family, validity with overlap power $c$ requires $c\le p'$, where $p'=p/(p-1)$.
• The three-function case has earlier bounds of Carlen, Frank, and Lieb, improved in the 2026 work.

June 2026 claimed all-real resolution

A June 2026 preprint constructs counterexamples disproving power $2$ for every $p>2$ and proves power $p'$ for every integer $p\ge2$. A later June 2026 paper claims, in a remark, a commutative $L^p$ proof of the optimal power $p'$ for every real $p\ge2$; this is not independently verified here. The sources credit Grok with substantial assistance, including the counterexample structure and parts of the proof.

Current status (as of September 2026): Power $2$ is disproved for every $p>2$, and power $p'$ is established for integer $p\ge2$; a claimed extension to every real $p\ge2$ remains unverified.