On $k$-convex hulls

For every integer $k\ge 2$ and every $R>1$ one can find a dimension $n$ and construct a symmetric convex body $K\subset {\mathbb{R}}^n$ with $\text{diam}{Q}_{k-1}(K)\ge R\cdot \text{diam}{Q}_k(K)$, where $Q_k(K)$ denotes the $k$-convex hull of $K$. The purpose of this short note is to show that this result due to E.\ Kopecká is impossible to obtain if one additionally requires that all isometric images of $K$ satisfy the same inequality. To this end, we introduce the dual construction to the $k$-convex hull of $K$, which we call the $k$-cross approximation of $K$. We also prove an infinite-dimensional version of the main result that holds in general Hilbert spaces.