  We investigate renormalized curvature flow (RCF) and stochastic renormalized
curvature flow (SRCF) for convex sets in the plane.RCF is the gradient descent
flow for logarithm of $\sigma/\lambda^2$ where $\sigma$ is the perimeter and
$\lambda$ is the volume. SRCF is RCF perturbated by a Brownian noise and has
the remarkable property that it can be intertwined with the Brownian motion,
yielding a generalization of Pitman "2M-X" theorem. We prove that along RCF,
entropy $\mathcal{E}_t$ for curvature as well as $h_t:=\sigma_t/\lambda_t$ are
non-increasing. We deduce infinite lifetime and convergence to a disk after
normalization.For SRCF the situation is more complicated. The process $(h_t)_t$
is always a supermartingale. For $(\mathcal{E}_t)_t$ to be a supermartingale,
we need that the starting set is invariant by the isometry group $G_n$
generated by the reflection with respect to the vertical line and the rotation
of angle $2\pi/n$ with $n\ge 3$. But for proving infinite lifetime, we need
invariance of the starting set by $G_n$ with $n\ge 7$. We provide the first
SRCF with infinite lifetime which cannot be reduced to a finite dimensional
flow. Gage inequality plays a major role in our study of the regularity of
flows, as well as a careful investigation of morphological skeletons. We
characterize symmetric convex sets with star shaped skeletons in terms of
properties of their Gauss map. Finally, we establish a new isoperimetric
estimate for these sets, of order $1/n^4$ where $n$ is the number of branches
of the skeleton.
