I was studying the paper “An improved bound for the Shapley-Folkman theorem” (https://doi.org/10.1016/j.jmateco.2020.04.003), and had trouble visualizing these concepts (on the first page of the paper):
(For any subset $X$ of $\mathbb{R}^m$, $\operatorname{co}X$ denotes its convex hull).
For any nonempty subset $S$ of $\mathbb{R}^m$, define the diameter of $S$ by $$\operatorname{diam}(S):=\sup_{x,y{\in}S}\Vert{x-y}\Vert$$ define the radius of $S$ by $$\operatorname{rad}(S):=\inf_{y{\in}\mathbb{R}^m}\sup_{x{\in}S}\Vert{x-y}\Vert$$ define the inner diameter of $S$ by $$\operatorname{indiam}(S):=\sup_{y{\in}\operatorname{co}S}\inf_{T{\subseteq}S:y{\in}\operatorname{co}T}\operatorname{diam}(T)$$ and define the inner radius of $S$ by $$\operatorname{inrad}(S):=\sup_{y{\in}\operatorname{co}S}\inf_{T{\subseteq}S:y{\in}\operatorname{co}T}\operatorname{rad}(T)$$
(There seems to be a visualization of this on Wikipedia, but I don’t completely understand it: https://en.wikipedia.org/wiki/Shapley%E2%80%93Folkman_lemma#Shapley%E2%80%93Folkman_theorem_and_Starr's_corollary)
I understand that $\operatorname{diam}(S)$ is the maximum distance between any two arbitrary points $x$ and $y$ in a set. This is easy enough to visualize. However, it is quite tough for me to understand what $\operatorname{rad}(S)$, $\operatorname{indiam}(S)$, and $\operatorname{inrad}(S)$ mean, and to visualize them. Can I get some help on this?