  This article introduces a novel method for the implementation of shape
optimisation with Lipschitz domains. We propose to use the shape derivative to
determine deformation fields which represent steepest descent directions of the
shape functional in the $W^{1,\infty}-$ topology. The idea of our approach is
demonstrated for shape optimisation of $n$-dimensional star-shaped domains,
which we represent as functions defined on the unit $(n-1)$-sphere. In this
setting we provide the specific form of the shape derivative and prove the
existence of solutions to the underlying shape optimisation problem. Moreover,
we show the existence of a direction of steepest descent in the $W^{1,\infty}-$
topology. We also note that shape optimisation in this context is closely
related to the $\infty-$Laplacian, and to optimal transport, where we highlight
the latter in the numerics section. We present several numerical experiments
illustrating that our approach seems to be superior over existing Hilbert space
methods, in particular in developing optimal shapes with corners.
