\(n\)-cubes inscribed in simplices (Q1775250)

On each side of a given triangle, there is a unique inscribed square that rests on that side, i.e., a square having two vertices that lie on that side (or its extension) and having the other two vertices lie on the remaining sides of the triangle. (For constructing such squares, see Problem 18 of Part I of \textit{G. Polya's} book [How to solve it (Princeton University Press, N. J.) (1945; Zbl 0061.00616)]). The interesting question of which of the three inscribed squares is the smallest is answered by \textit{H. Bailey and D. Detemple} in [Math. Mag. 71, 278-284 (1998)], where they prove that if \(AB > BC\), then the inscribed square of \(ABC\) that rests on \(AB\) is smaller than the one that rests on \(BC\). In the paper under review, the author generalizes this result to \(n\)-simplices for all \(n\), considerately separating the case \(n=3\) for easier reading. Specifically, an \(n\)-cube \(C\) is defined, inductively, to be inscribed in an \(n\)-simplex \(S\) if a face \(F\) of \(C\) lies in the hyperplane determined by a facet of \(S\), and if the face \(F'\) of \(C\) opposite to \(F\) is, as an \((n-1)\)-cube, inscribed in the \((n-1)\)-simplex formed by intersecting \(S\) with the hyperplane determined by \(F'\). Associated to each of the \(\frac{(n+1)!}{2}\) \(n\)-cubes inscribed in an \(n\)-simplex \(S\) is a sequence \(H_1 \subset H_2 \subset \cdots \subset H_{n-1}\), where \(H_j\) is a \(j\)-face of \(S\). These are the faces of \(S\) where the \textit{resting} faces of \(C\) (of various dimensions) rest. Among other things, the author proves that if \((H_1, \dots,H_{n-1})\) and \((H_1^{\prime}, \dots,H_{n-1}^{\prime})\) are the sequences associated to the inscribed \(n\)-cubes \(C\) and \(C^{\prime}\), and if \(\mu_j(H_j) \geq \mu_j(H_j^{\prime})\) for all \(j\), then \(\mu_n(C) \leq \mu_n(C^{\prime})\). Here \(\mu_j\) stands for the \(j\)-dimensional volume or Lebesgue measure.