  For hypocoercive linear kinetic equations we first formulate an optimisation
problem on a spatially dependent jump rate in order to find the fastest decay
rate of perturbations. In the Goldstein-Taylor model we show (i) that for a
locally optimal jump rate the spectral gap is determined by multiple, possible
degenerate, eigenvectors and (ii) that globally the fastest decay is obtained
with a spatially homogeneous jump rate. Our proofs rely on a connection to
damped wave equations and a relationship to the spectral theory of
Schr{\"o}dinger operators.
