  The classical isoperimetric inequality can be extended to a general normed
plane. In the Euclidean plane, the defect in the isoperimetric inequality can
be calculated in terms of the signed areas of some singular sets. In this paper
we consider normed planes with smooth by parts unit balls and the corresponding
class of admissible curves. For such an admissible curve, the singular sets are
defined as projections in the subspaces of symmetric and constant width
admissible curves. In this context, we obtain some improved isoperimetric
inequalities whose equality hold for symmetric or constant width curves.
