We investigate renormalized mean curvature flow (RMCF) and stochastic renormalized mean curvature flow (SRMCF) for convex sets in the plane. RMCF is the inverse gradient flow for logarithm of square of the perimeter divided by the volume. SRMCF is RMCF perturbated by some Brownian noise and has the remarkable property that it can be intertwined with Brownian motion, yielding a generalization of Pitman "2M-X" theorem. We prove that along RMCF, entropy E_t for curvature as well as the quotient h_t of perimeter divided by the volume are non-increasing. We deduce infinite lifetime and convergence to a disk after normalization. For SRMCF the situation is more complicated. As h_t is always a supermartingale, for E_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 2pi/n, for some n larger than or equal to 3. But for proving infinite lifetime, we need invariance of the starting set by G_n for some n larger than or equal to 7. We provide the first SRMCF 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.