Errata for "Cutting corners"

"On occasion, we also need to talk about individual midpointed sets \( C \Subset G \), meaning that the family \( \{C\} \) is (\(S\)-)midpointed"

This definition should say: A set \(C\) is \(S\)-midpointed if, whenever \(gh, gh^{-1} \in C\) for \(h \in S\), then also \(g \in C\).

One can interpret the definition this way, from what is written in the paper: If we consider a singleton family of convex sets, then according to the explanation below Definition 3.1, \( \overline{A} \) should be interpreted as the intersection of convex sets containing it. When the sets do not cover the group, it is natural to interpret an empty intersection to equal \(G\) (or any larger set). Then \(g \in \overline{\{gh, gh^{-1}\}}\) in the definition of a midpointed family indeed means \(g \in C\) when \(gh, gh^{-1} \in C\), and holds trivially otherwise, so this indeed gives the correct meaning.

I don't see a reasonable interpretation such that the definition ends up meaning something else, but it is natural to interpret the definition as being meaningless, since we did not address empty intersections explicitly, so I added this clarification.

Last update: 16.6.2023