The multidimensional generalized Hegselmann–Krause conjecture

Let $m\geq 1$, and consider opinions in $\mathbb{R}^m$ with a norm replacing the absolute value in the update formula. Let $\Omega$ be an arbitrary generalized weighted asynchronous bounded-confidence opinion system (WASBOCOS) with structural parameters $\varepsilon$, $\alpha$, $\beta$, and $n$, and let $\gamma>0$. The truth seekers are $\phi(\Omega,\gamma)$-fold interrupted convergent in these parameters if their convergence to the truth can be divided into at most $\phi(\Omega,\gamma)$ interruptions before eventually remaining within distance $\gamma$ of the truth. Multidimensional generalized Hegselmann–Krause conjecture. The $m$-dimensional generalized Hegselmann–Krause conjecture holds, and there exists a function $\phi(\Omega,\gamma)$ such that the truth seekers in every generalized WASBOCOS $\Omega$ are $\phi(\Omega,\gamma)$-fold interrupted convergent in $\varepsilon$, $\alpha$, $\beta$, and $n$. The claim extends the one-dimensional convergence problem to vector-valued opinions and asserts a stronger uniform interruption bound; the source states that proving it by the paper’s approach would require new ideas and tools, and gives no resolution status.