  We study the dynamic evolution of geometric structures in a poly-degenerate
system represented by a $q$-state Potts model with non-conserved order
parameter that is quenched from its disordered into its ordered phase. The
numerical results obtained with Monte Carlo simulations show a strong relation
between the statistical properties of hull perimeters in the initial state and
during coarsening: the statistics and morphology of the structures that are
larger than the averaged ones are those of the initial state while the ones of
small structures are determined by the curvature driven dynamic process. We
link the hull properties to the ones of the areas they enclose. We analyze the
linear von-Neumann--Mullins law, both for individual domains and on the
average, concluding that its validity, for the later case, is limited to
domains with number of sides around 6, while presenting stronger violations in
the former case.
