Regular simplices passing through holes (Q964186)

Denote by \(S_n\), \(Q_n\) and \(B_n\) the \(n\)-dimensional simplex, hypercube and ball of unit diameter, respectively. An \((n-1)\)-dimensional convex body \(\Theta\), in \(\mathbb{R}^n\), with \(\text{diam}(\Theta)=1\) is called a hole-shape. For a hole-shape \(\Theta\), the authors investigate two types of minimum diameters \(\gamma\) and \(\Gamma\) defined by: \[ \begin{aligned}\gamma(n,\Theta)&:= \min\{d: S_n \;\text{can pass through the hole}\; d\Theta\},\\ \Gamma (n,\Theta)&:= \min\{d: S_n \subset (d\Theta)\times \mathbb{R}\}.\end{aligned} \] Among other results the authors show that: (i) \(\lim_{n\rightarrow \infty} \gamma(n, S_{n-1})=\lim_{n\rightarrow \infty} \Gamma(n, S_{n-1})=1\), (ii) \(\lim_{n\rightarrow \infty} \gamma(n, B_{n-1})=3\sqrt{2}/4\), \(\lim_{n\rightarrow \infty} \Gamma(n, B_{n-1})=\sqrt{2}\), and (iii) \(3\sqrt{2}/4\leq \lim_{n\rightarrow \infty} \gamma(n, Q_{n-1})\leq 2\), \(\sqrt{2}\leq \lim_{n\rightarrow \infty} \Gamma(n, Q_{n-1})\leq 2.\)