Let $f$ be a positive function over the unit sphere $S^{d-1}$. Minkowski's problem is to find a convex body $K$ in ${\mathbb R}^d$, whose Gauss curvature is prescribed as a function of the normal direction: $\kappa=f(\vec n)$. An "obvious" necessary condition is the vector-valued identity $$\int_{S^{d-1}}f(\omega)\omega ds(\omega)=0.$$ Pogorelov proved that this condition (plus some modest regularity) is actually sufficient. He presented the complete proof in a book published in 1978 by Wiley & Sons.

Is there a close formula for the volume of $K$ in terms of $f$ ? If not, are there nice estimates of this volume in terms of $f$ ?

mighthelp, but I have not penetrated enough to be confident that it is relevant. Alexandrov, Victor, Natalia Kopteva, and S. S. Kutateladze. "Blaschke addition and convex polyhedra." arXiv preprint math/0502345 (2005). $\endgroup$ – Joseph O'Rourke Jan 11 at 14:10