On a Simplex Contained in a Ball

Let Bₙ be the n-dimensional unit ball given by the inequality \|x\|≤ 1, where $\|x\|$ is the standard Euclid norm in Rⁿ. For an n-dimensional nondegenerate simplex S, we denote by E the ellipsoid of minimum volume which contains S. Suppose S⊂ Bₙ, 0≤ m≤ n-1. Let G be any m-dimensional face of S and let H be the opposite $(n-m-1)$-dimensional face. Denote by g and h the centers of gravity of G and H respectively. Define y as the intersection point of the line passing from g to h with the boundary of E. Let us call the face G suitable if y∈ Bₙ. Earlier it was proved that each simplex S⊂ Bₙ has a suitable face of any dimension ≤ n-1. We show the following. If some vertex of S is suitable, then there exists a suitable face of any dimension ≤ n-1 which contains this vertex.