Simplexes with a strictly monotone entropy number sequence (Q1200961)

Let \(S\) denote the simplex \(\{x:\;x\geq 0,\;\langle a,x\rangle\leq b\}\) (\(a\in\mathbb{R}_ +^ n\), \(b>0\)) in \(\ell_ \infty^ n\), the \(n\)- dimensional space with the maximum norm. It is shown that the numbers \(\varepsilon_ k(S)=\min\{\varepsilon>0\): \(\exists k\) closed \(\varepsilon\)-balls covering \(S\}\) are strictly decreasing if the coordinates of \(a\) are rationally independent.