Sufficiency of simplex inequalities

Let z 0, …, zn be the (n − 1)-dimensional volumes of facets of an n-simplex. Then we have the simplex inequalities: zp < z 0+ … + ž p + … +zn (0 ≤ p ≤ n), generalizations of the triangle inequalities. Conversely, suppose that numbers z 0, …, zn > 0 satisfy these inequalities. Does there exist...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 2016-03, Vol.144 (3), p.1299-1307
Main Author: Izumi, Shuzo
Format: Article
Language:English
Subjects:
D. GEOMETRY
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Summary:Let z 0, …, zn be the (n − 1)-dimensional volumes of facets of an n-simplex. Then we have the simplex inequalities: zp < z 0+ … + ž p + … +zn (0 ≤ p ≤ n), generalizations of the triangle inequalities. Conversely, suppose that numbers z 0, …, zn > 0 satisfy these inequalities. Does there exist an n-simplex the volumes of whose facets are them? Kakeya solved this problem affirmatively in the case n = 3 and conjectured the assertion for all n ≥ 4. We prove that his conjecture is affirmative. 2010 Mathematics Subject Classification. Primary 51M16.
ISSN:0002-9939
1088-6826
DOI:10.1090/proc12756