On $k$-convex hulls
Davide Ravasini
TL;DR
The paper investigates $k$-convex hulls $Q_k(K)$ of symmetric convex bodies and their dual cross-approximations $R_k(K)$, establishing a rotation-robust bound that $Q_{k-1}(UK)\subseteq 3\,Q_k(UK)$ holds for some $U\in SO(n)$ (and a $(3+\delta)$-type bound in Hilbert spaces). The approach centers on the duality $Q_k(K)^{\circ}=R_k(K^{\circ})$ and a constructive, greedy basis selection to produce the required inclusions, with a finite-dimensional argument extended to infinite dimensions via transfinite induction. The main contributions include a dimension-free constant (namely 3) for finite-dimensional spaces and a parallel infinite-dimensional generalization, together with a detailed polar-dual framework that clarifies the relationship between $k$-convex hulls and $k$-cross approximations. These results show that Kopecká’s phenomenon cannot be uniformly sharp across all rotations and illuminate dual structures and alternative cross-approximation notions in both finite and infinite-dimensional settings.
Abstract
For every integer $k\geq 2$ and every $R>1$ one can find a dimension $n$ and construct a symmetric convex body $K\subset\mathbb{R}^n$ with $\text{diam}\,Q_{k-1}(K)\geq R\cdot\text{diam}\,Q_k(K)$, 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.