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

A spectral isoperimetric inequality for cones