Timeline for About Minkowski's problem
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7 events
| when toggle format | what | by | license | comment | |
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| Jan 11, 2019 at 16:37 | comment | added | Ivan Izmestiev | 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. | |
| Jan 11, 2019 at 16:34 | comment | added | Ivan Izmestiev | 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 $f$. | |
| Jan 11, 2019 at 15:13 | comment | added | Deane Yang | 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 $$ \int_{S^{d-1}} \omega\,dS(\omega) = 0$$ | |
| Jan 11, 2019 at 15:12 | comment | added | Paata Ivanishvili | There is an inequality which relates the volume of the convex body, Gauss curvature and the surface area measure of the convex body $\left( \frac{as(K)}{as(B^{n})}\right)^{\frac{n+1}{n-1}} \leq \frac{vol(K)}{vol(B^{n})}$, where $B^{n}$ is n dimensional unit ball, and $as(K) = \int_{\partial K} f(x)^{\frac{1}{n+1}}d\mu_{\partial K}(x)$, here $d\mu_{\partial K}(x)$ is a surface area measure. This inequality becomes equality for ellipsoids and it is equivalent to Blashke--Santalo inequality. See page 2 Grote--Werner, Approximation of smooth convex bodies by random polytopes | |
| Jan 11, 2019 at 15:08 | comment | added | Deane Yang | 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 $K$. | |
| Jan 11, 2019 at 14:10 | comment | added | Joseph O'Rourke | 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). | |
| Jan 11, 2019 at 13:59 | history | asked | Denis Serre | CC BY-SA 4.0 |