Convex bodies passing through holes

For a given convex body, find a “small” wall hole through which the convex body can pass. This type of problems goes back to Zindler [14] in 1920, who considered a convex polytope which can pass through a fairly small circular holes. A related topic known as Prince Rupert’s problem can be found in [2]. Here we concentrate on the case when the convex body is a regular tetrahedron or a regular n-simplex. For a compact convex body K⊂Rn, let diam(K) and width(K) denote the diameter and width of K, respectively. For d > 0 let dK denote the convex body with diameter d and homothetic to K. Let Sn, Qn, and Bn denote the n-dimensional regular simplex, the n-dimensional hypercube, and the n-dimensional ball, respectively. Thus, 1Sn has side length 1, 1Qn has side length 1/ √ n, and 1Bn has radius 1/2. Let H ⊂ Rn−1 be a convex body, which we will call a hole. Let Π be the hyperplane containing H, which divides Rn into Π and two (open) half spaces Π+ and Π−. We want to push 1Sn from Π+ to Π− through H. In this situation, we are interested in two types of “small” holes, namely, γ(n,H) := min{d : 1Sn can pass through the hole of dH in Rn},