On $k$-convex hulls

TL;DR

The paper demonstrates that a certain geometric inequality involving $k$-convex hulls cannot be uniformly satisfied by all isometric images of a convex body, introducing the $k$-cross approximation and extending results to infinite-dimensional Hilbert spaces.

Contribution

It shows the impossibility of uniform inequalities for all isometric images of convex bodies and introduces the dual concept of $k$-cross approximation.

Findings

The inequality involving $k$-convex hulls is not uniformly valid for all isometric images.

Introduction of the $k$-cross approximation as a dual construction.

Extension of the main geometric result to infinite-dimensional Hilbert spaces.

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.\ Kopeck\'{a} 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.