Sufficiency of simplex inequalities (Q2790206)

The paper under review proves that if \(z_0 \geq z_1 \geq \dots \geq z_n\) are given positive numbers, then there exists an \(n\)-dimensional simplex \(S\) such that \(z_0, z_1, \dots, z_n\) are the contents (i.e., the \((n-1)\)-dimensional volumes) of the facets (i.e., the \((n-1)\)-dimensional faces) of \(S\) if and only if \(z_1 + \dots + z_n > z_0\). This generalizes Propositions 20 and 22 of Book I of Euclid's \textit{Elements} which, when combined, state that the positive numbers \(x \geq y \geq z\) can serve as the side lengths of a triangle if and only if \(y+z > x\).NEWLINENEWLINEThe author of the paper under review is apparently unaware that the same result has already been proved (as Theorem 5.1) by \textit{L. Gerber} [Pac. J. Math. 56, 97--111 (1975; Zbl 0303.52004)].