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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 $f$ ?

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    $\begingroup$ This might help, 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
    Commented Jan 11, 2019 at 14:10
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    $\begingroup$ The function $f$ is the reciprocal of Gauss curvature, $f = \kappa^{-1}$. I do not know of any closed form formula for the volume of the solution $K$. The volume of $K$ is given by $$ V = \frac{1}{d}\int_{S^{d-1}} h(\omega)f(\omega)\,ds(\omega), $$ where $ds$ is the standard $(d-1)$-volume measure on $S^{d-1}$ and $h$ is the support function of $K$. $\endgroup$
    – Deane Yang
    Commented Jan 11, 2019 at 15:08
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    $\begingroup$ By the way, no regularity assumptions are needed at all. The integrand $f\,ds$ is well defined as a measure $dS$ on $S^{n-1}$ (called the surface area measure) for any convex body. Minkowski solved the Minkowski problem for polytopes, where the measure is a finite discrete measure. Alexandrov and, independently Fenchel-Jessen, extended Minkowski's solution to arbitrary Borel measures on the sphere such that the measure is not supported on a hypersphere and satisfies $$ \int_{S^{d-1}} \omega\,dS(\omega) = 0$$ $\endgroup$
    – Deane Yang
    Commented Jan 11, 2019 at 15:13
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    $\begingroup$ The function $f$ is the determinant of the Hessian of the support function $h$, and the volume can be computed from $h$ by the formula given in the first comment of Deane Yang. So, it is unlikely that there is an explicit formula for the volume in terms of $f$. $\endgroup$
    – Ivan Izmestiev
    Commented Jan 11, 2019 at 16:34
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    $\begingroup$ Also, in the discrete case the function $f$ gives the areas of facets (faces of codimension $1$) with prescribed directions of normals. One could compute the volume if one knew the heights of all facets. But I am quite sure that there is no explicit formula for the heights. $\endgroup$
    – Ivan Izmestiev
    Commented Jan 11, 2019 at 16:37

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