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Sufficiency of simplex inequalities


Author: Shuzo Izumi
Journal: Proc. Amer. Math. Soc. 144 (2016), 1299-1307
MSC (2010): Primary 51M16
DOI: https://doi.org/10.1090/proc12756
Published electronically: July 8, 2015
MathSciNet review: 3447680
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Abstract: Let $ z_0,\dots ,z_n$ be the $ (n-1)$-dimensional volumes of facets of an $ n$-simplex. Then we have the simplex inequalities: $ z_p < z_0+\dots +\check {z}_p+\dots +z_n$ $ (0\le p\le n)$, generalizations of the triangle inequalities. Conversely, suppose that numbers $ z_0,\dots ,z_n>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\ge 4$. We prove that his conjecture is affirmative.


References [Enhancements On Off] (What's this?)

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Additional Information

Shuzo Izumi
Affiliation: Research Center of Quantum Computing, Kinki University, Higashi-Osaka 577-8502, Japan
Email: sizmsizm@gmail.com

DOI: https://doi.org/10.1090/proc12756
Received by editor(s): January 23, 2014
Received by editor(s) in revised form: February 6, 2015
Published electronically: July 8, 2015
Communicated by: Michael Wolf
Article copyright: © Copyright 2015 American Mathematical Society