The set of geometric medians for four-element subsets in Lindenstrauss spaces (Q785911)

A Banach space \(X\) is a Lindenstrauss space if \(X^*\) is isometrically isomorphic to an \(L_1(\mu)\) space for some measure \(\mu\). (There appears to be a misprint in Theorem A.(3) in the paper -- the word isometrically is missing.) For a Banach space \(X\) and a set of points \(M_n = \{x_1,\ldots,x_n\}\) in \(X\) consider \[ |st|(M_n) :=\inf \left\{\sum_{k=1}^n \|x_k - x\| : x \in X\right\}. \] The Steiner points or geometric medians of \(M_n\) form the set \[ st(M_n)=\left\{s \in X:\sum_{k=1}^n \|x_k - s\| = |st|(M_n)\right\}. \] The paper under review studies Steiner points for four-point subsets \(M_4\) in Lindenstrauss spaces. In particular, the connection between \(st(M_4)\) and \(st(M)\) where \(M\) is a three-point subset of \(M_4\) is considered. In the case where \(X = C(K)\), for some compact Hausdorff space \(K\), it is shown that the multivalued map \(S:(x_1,\dots,x_4) \to st((x_1,\dots,x_4))\) has a Lipschitz selection.