The Hilton-Milner theorem in the ${\mathrm{\ell }}_2$-norm

The Hilton-Milner theorem is a classical theorem in extremal set theory which determines the maximum size of a nontrivial $k$-uniform intersecting family. In this paper, we prove an exact version of the Hilton-Milner theorem in the ${\mathrm{\ell }}_2$-norm. Our method also gives an asymptotic version of the Hilton-Milner theorem in the $(j,p)$-norm.