Geometric evolution of convex surfaces to a round sphere Let $ = ^2$ be an embedded convex surface in $\mathbb R^3$ and let $ ∶  → ^2$ be the Gauss map for $.$ Let $_$ be the area measure on $$ and $_∗_$ the corresponding pushforward measure on the sphere, i.e., the measure assigning mass $_ (^{−1}())$ to a subset $ \subset \mathbb S^2.$ We may evolve $_∗_$ by the heat equation on $\mathbb S^2.$ Does the solution to Minkowski’s problem imply that the resulting heat flow corresponds to a 1-parameter family of embedded convex surfaces, and if so can we provide a better description of them?
 A: This is a little long for a comment.  I also haven't double checked things so may have made mistakes.
I'm going to treat the case of a convex curve in the plane (I imagine something similar works for surfaces).  I'm also going to assume the unit normal, $N$, is a diffeomorphism onto the circle.  In this case we have $N(p)=(\cos(\theta(p)), \sin(\theta(p)))$ where $$\theta:M\to \mathbb{R}/2\pi \mathbb{Z}=\mathbb{S}^1.$$  We also think of $\theta$ as a coordinate on $\mathbb{S}^1$ (the unit circle).
In this case, I think we have that
$$
N_*V_M =\frac{1}{\kappa(N^{-1}(\theta))} d\theta,
$$
where here $\kappa$ is the geodesic curvature of the curve $M$.
So you are proposing to evolve the geometric quantity  $\psi=\frac{1}{\kappa(N^{-1}(\theta))}=\frac{1}{\hat{\kappa}(\theta)}$ by the heat equation.
In general. a function $\hat{\kappa}:\mathbb{S}^1\to \mathbb{R}$ that is positive is the curvature of a closed convex curve  (i.e. is the $\hat{\kappa}$ form above for some closed convex curve) if and only if
$$
\int_0^{2\pi}(\cos\theta, \sin \theta)\frac{1}{\hat{\kappa}(\theta)} d\theta=(0,0).
$$
This fact was observed (in the context of geometric heat flows) by Gage-Hamilton (in "The heat equation shrinking convex plane curves").
In particular, it seems that this condition is preserved by your proposed heat equation.
One final remark.  If we let
$$u(\theta)=\mathbf{x}(N^{-1}(\theta))\cdot N(\theta)$$
be the support function of the curve then it should be the case that
$$
u_{\theta\theta}(\theta)+u(\theta)=\frac{1}{\kappa(N^{-1}(\theta))}.
$$
In particular, evolving the support function by the heat equation seems to induce the same evolution as what you are proposing (in fact if you impose the condition on the curvature from above that makes sure everything comes from a closed curve then it seems like one can also go the other direction due to the invertibility of the operator in the appropriate subspace this condition defines).  The flow of the support function by the linear heat equation maybe something that has been studied before (though I don't know off the top of my head).
EDIT:  I did some more searching and the following paper paper by Chow, Liou and Tsai, considers a linear heat flow on the support function of a convex hypersurface and relates it to the (inverse) harmonic mean curvature flow.  Perhaps this gives some ideas or at least references you can look at further.
