  We study downward deviations of the boundary of the range of a transient walk
on the Euclidean lattice. We describe the optimal strategy adopted by the walk
in order to shrink the boundary of its range. The technics we develop apply
equally well to the range, and provide pathwise statements for the {\it Swiss
cheese} picture of Bolthausen, van den Berg and den Hollander \cite{BBH}.
