  In this note we investigate three-dimensional Schr\"odinger operators with
$\delta$-interactions supported on $C^2$-smooth cones, both finite and
infinite. Our main results concern a Faber-Krahn-type inequality for the
principal eigenvalue of these operators. The proofs rely on the
Birman-Schwinger principle and on the fact that circles are unique minimizers
for a class of energy functionals. The main novel idea consists in the way of
constructing test functions for the Birman-Schwinger principle.
