Cerenkov-effect in other dimensions

\(3+1\) dimensional spacetimes are special because the Huygens-principle is perfectly fulfilled as spherical waves are perfectly expanding shells, or in other words, as all partial waves travel at the same speed \(c\). Would there be a Cerenkov effect in materials in \(2+1\) dimensions, the issue being that there are partial waves travelling inside the light cone, or in other words: particles, no matter how slow they are, are always faster than some of the partial waves.