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On the Aleksandrov-Fenchel inequality. (English) Zbl 0567.52004

Discrete geometry and convexity, Proc. Conf., New York 1982, Ann. N.Y. Acad. Sci. 440, 132-141 (1985).
[For the entire collection see Zbl 0564.00011.]
For convex bodies \(K_ 1,...,K_ n\) in n-dimensional Euclidean space, the Aleksandrov-Fenchel inequality says \(V(K_ 1,K_ 2,K_ 3,...,K_ n)^ 2\geq V(K_ 1,K_ 1,K_ 3,...,K_ n)\cdot V(K_ 2,K_ 2,K_ 3,...,K_ n)\), where V denotes the mixed volume.
The author states three conjectures concerning equality in the above inequality, summarizes which special parts of his conjectures are known to be true, and gives additional results supporting them. He thus hopes to inspire further research in order to settle the case of equality.
Reviewer: B.Kind

MSC:

52A40 Inequalities and extremum problems involving convexity in convex geometry

Citations:

Zbl 0564.00011