  L\'evy walks are random walk processes whose step-lengths follow a
long-tailed power-law distribution. Due to their abundance as movement patterns
of biological organisms, significant theoretical efforts have been devoted to
identifying the foraging circumstances that would make such patterns
advantageous. However, despite extensive research, there is currently no
mathematical proof indicating that L\'evy walks are, in any manner, preferable
strategies in higher dimensions than one. Here we prove that in finite
two-dimensional terrains, the inverse-square L\'evy walk strategy is extremely
efficient at finding sparse targets of arbitrary size and shape. Moreover, this
holds even under the weak model of intermittent detection. Conversely, any
other intermittent L\'evy walk fails to efficiently find either large targets
or small ones. Our results shed new light on the \emph{L\'evy foraging
hypothesis}, and are thus expected to impact future experiments on animals
performing L\'evy walks.
