The Minkowski problem for polytopes.(English)Zbl 1053.52015
The Minkowski problem for polytopes concerns the existence and uniqueness, up to translations, of a convex polytope with given normal vectors and areas of its facets. The classical approach first settles the existence by solving an extremal problem and then establishes the essential uniqueness by employing a fundamental result of the Brunn-Minkowski theory of mixed volumes, namely Minkowski’s first inequality and its equality condition.
In the present paper, the author gives a proof by induction with respect to the dimension which yields simultaneously the solution of the Minkowski problem and the Minkowski inequality with equality condition (for polytopes).
In the present paper, the author gives a proof by induction with respect to the dimension which yields simultaneously the solution of the Minkowski problem and the Minkowski inequality with equality condition (for polytopes).
Reviewer: Rolf Schneider (Freiburg i. Br.)
MSC:
| 52B11 | \(n\)-dimensional polytopes |
| 52A40 | Inequalities and extremum problems involving convexity in convex geometry |
| 52A39 | Mixed volumes and related topics in convex geometry |
Keywords:
Minkowski problem;Minkowski’s inequality;mixed volume;polytope;isoperimetric;geometric inequalityCite
Full Text:DOI
References:
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