  This paper investigates geometric properties and well-posedness of a mean
curvature flow with volume-dependent forcing. With the class of forcing which
bounds the volume of the evolving set away from zero and infinity, we show that
a strong version of star-shapedness is preserved over time. More precisely, it
is shown that the flow preserves the $\rho$-reflection property, which
corresponds to a quantitative Lipschitz property of the set with respect to the
nearest ball. Based on this property we show that the problem is well-posed and
its solutions starting with $\rho$-reflection property become instantly smooth.
Lastly, for a model problem, we will discuss the flow's exponential convergence
to the unique equilibrium in Hausdorff topology. For the analysis, we adopt the
approach developed in [Kim-Feldman, 2014] to combine viscosity solutions
approach and variational method. The main challenge lies in the lack of
comparison principle, which accompanies forcing terms that penalize small
volume.
