Fixed-particle-number optimizers for the Lieb--Oxford inequality

Abstract

Let $\mathsf{d}\geq1$, $0<\mathsf{s}<\mathsf{d}$, and $N\geq1$. We prove that the optimal fixed-particle-number constant $\Lambda_N(\mathsf{s},\mathsf{d})$ in the Riesz Lieb--Oxford inequality is attained and that these constants are strictly increasing in $N$. The proof combines grand-canonical concentration--compactness with a strict one-particle extension. After recentering, a limiting plan arising from a maximizing sequence may assign positive probability to several particle numbers and hence be grand-canonical. A strict $N$-particle completion excludes this case, while the inequalities $\Lambda_N>\Lambda_k$ for $k \Lambda_N(\mathsf{s},\mathsf{d})$. Together, these implications close an induction beginning at $N=1$.