On affine-regular cross-polyhedra inscribed in a convex body

Let $A$ be an affine-regular cross-polyhedron, i.e., the convex hull of $n$ segments $A_1B_1,\dots,A_nB_n$ in $\mathbb R^n$ that have common midpoint $O$ and do not lie in a hyperplane. The affine flag $F(A)$ of $A$ is defined as the chain $O\in L_1\subset\dots\subset L_n=\mathbb R^n$, where $L_k$ is the affine hull of the segments $A_1B_1,\dots,A_kB_k$. It is proved that each convex body $K\subset\mathbb R^n$ is circumscribed about an affine-regular cross-polyhedron $A$ such that the flag $F(A)$ satisfies the following condition for each $k\in\{2,\dots,n\}$: $(k-1)$-planes of support at $A_k$ and $B_k$ for the body $L_k\cap K$ in the $k$-plane $L_k$ are parallel to $L_{k-1}$. Each such $A$ has volume at least $V(K)/2^{n(n-1)/2}$. Bibl. – 4 titles.