  Diffusing diffusivity models, polymers in the grand canonical ensemble and
polydisperse, and continuous-time random walks all exhibit stages of
non-Gaussian diffusion. Is non-Gaussian targeting more efficient than Gaussian?
We address this question, central to, e.g., diffusion-limited reactions and
some biological processes, through a general approach that makes use of
Jensen's inequality and that encompasses all these systems. In terms of
customary mean first-passage time, we show that Gaussian searches are more
effective than non-Gaussian ones. A companion paper argues that non-Gaussianity
becomes instead highly more efficient in applications where only a small
fraction of tracers is required to reach the target.
