Estimates for the Jung constant in Banach lattices (Q1976612)

Given a Banach space \(E\), the Jung constant, \(J(E)\) is related to the geometry of \(E\), specifically, \(J(E)= \sup\{2r(A): A\subset E, d(A)\leq 1\}\), where ``\(d\)'' and ``\(r\)'' denote the diameter and radius. This paper provides sharp lower estimates for the Jung constant for a class of Banach lattices satisfying upper \(p\)-estimates and lower \(q\)-estimates for \(p\) and \(q\) in \([1,\infty)\). Specifically, \(E\) satisfies a lower \(q\)-estimate if there is a constant \(C>0\) so that \(\|\sum x_i\|\leq C(\sum\|x_i\|^q)^{1/q}\) for every finite set of disjointly supported elements \(\{x_1,x_2,\dots, x_n\}\). For example, it is shown that for \(E\) satisfying a lower \(q\)-estimate the value \(J(E)\) is greater than or equal to \(2^{1/q}\). The minimal value for the Jung constant among equivalent renormings is established for Lorentz spaces. Indeed, a variety of estimates and evaluations of this minimal value of the Jung constant are provided in several different settings. An estimate for the radius of a bounded sequence of disjointly supported functions in \(L_{p,\infty}\) is also established.