Every proper closed convex cone $K\subset\mathbb R^n$ with nonempty interior admits a logarithmically homogeneous self-concordant barrier $F:K^\circ\to\mathbb R$ with finite parameter $\nu$. For symmetric cones, including $\mathbb R^n_+$, second-order cones, and $\mathbb S^n_+$, the standard logarithmic barriers are self-concordant and have parameters equal to their ranks.