Jung constants of the \(\ell ^ n_ p\)-spaces
Let \(B_ 2\) be a subspace of a normed space \(B_ 1\), and let \(M\subset B_ 2\). Defne the Chebyshev radius \(r_{B_ 1}(M):=\inf_{x\in B_ 1}\sup_{y\in M}\| x-y\|_{B_ 2},\) and let \(d_{B_ 2}(M_ 2)\) be the diameter of M in \(B_ 2\). Define the Jung constant \(I(B_ 1,B_ 2)=\sup \{r_{B_ 1}(M)/d_{B_ 2}(M):\) \(M\subset B_ 2\}.\) In the paper there are given upper and lower estimates of Jung constants for finite dimensional spaces with mixed \(\ell_ p\)-norms in terms of the norm of a certain interpolation operator, and with a precise dependence on the dimension. This, in particular, yields an exact evaluation of Jung constants for infinite dimensional \(L_ p\)-spaces with an atomless measure.
upper and lower estimates of Jung constants for finite dimensional spaces with mixed \(\ell _ p\)-norms in terms of the norm of a certain interpolation operator
exact evaluation of Jung constants for infinite dimensional \(L_ p\)-spaces with an atomless measure
scientific article; zbMATH DE number 4200916
scientific article