  We analyze a gradient flow of closed planar curves minimizing the
anisoperimetric ratio. For such a flow the normal velocity is a function of the
anisotropic curvature and it also depends on the total interfacial energy and
enclosed area of the curve. In contrast to the gradient flow for the
isoperimetric ratio, we show there exist initial curves for which the enclosed
area is decreasing with respect to time. We also derive a mixed anisoperimetric
inequality for the product of total interfacial energies corresponding to
different anisotropy functions. Finally, we present several computational
examples illustrating theoretical results.
