# Smallest regular simplex containing the unit cube in $R^n$

What is the length $$e_n$$ of the edge of the smallest $$n$$-dimensional regular simplex $$S_n$$ containing the $$n$$-dimensional unit cube $$Q_n$$?

In particular, is there $$n$$ such that $$e_n<\sqrt{2}(n+1-\sqrt{n+1})$$?

The question is motivated by the question Two cubes in unit cube. A positive answer to the second part would (almost) imply that two cubes with side of length greater than $$1/2$$ would fit inside the unit cube without overlap. Where "almost" means "if the centers of the cube and the simplex are not too far away": the plan would be to cut the unit cube by a hyperplane orthogonal to a main diagonal and fit the small cube inside each of the two halves.

Since $$Q_n$$ contains a ball $$B_n$$ of diameter $$1$$, the simplex $$S_n$$ also contains $$B_n$$. Therefore, the altitude of $$S_n$$ is at least $$(n+1)/2$$ and so $$e_n \ge \frac{1}{\sqrt 2} \sqrt{n(n+1)}$$.

On the other hand, a "greedy" recursive construction (where a facet of $$Q_n$$ is contained in a facet of $$S_n$$) gives the upper bound $$e_n \le \sqrt{2}\left(\sqrt{\frac{1}{2}} + \sqrt{\frac{2}{3}} + \cdots +\sqrt{\frac{n}{n+1}}\right) = \sqrt{2}n - \Theta(\log n)$$.

The three-dimensional case hase been completely solved: Ogilvy and Robbins, 1976, Croft, 1980

It has been shown that the greedy construction is indeed optimal in $$\mathbb{R}^3$$; however, two other locally minimal regular tetrahedra are larger just by a few percent.

This paper by M. Nevskii is relevant. It is shown there that a simplex contains a translate of $[0,1]^n$ iff $\sum_i\frac1{d_i}\leq 1$, where $d_i$ is the length of the longest segment inside the simplex which is parallel to the $i$th axis. In particular, this clearly implies $e_n\geq n$ (and this is far from being sharp).