Sufficiency of simplex inequalities

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.