Some geometrical characterizations of $L_1$-predual spaces
Volume 270 / 2023
Abstract
Let $X$ be a real Banach space. For a non-empty finite subset $F$ and closed convex subset $V$ of $X$, we denote by ${\rm rad}_{X}(F)$, ${\rm rad}_{V}(F)$, ${\rm cent}_{X}(F)$ and $d(V, {\rm cent}_{X}(F))$ the Chebyshev radius of $F$ in $X$, the restricted Chebyshev radius of $F$ in $V$, the set of Chebyshev centers of $F$ in $X$ and the distance between the sets $V$ and ${\rm cent}_{X}(F)$ respectively. We prove that $X$ is an $L_{1}$-predual space if and only if for each four-point subset $F$ of $X$ and non-empty closed convex subset $V$ of $X$, \[{\rm rad}_{V}(F) = {\rm rad}_{X}(F) + d(V, {\rm cent}_{X}(F)).\] Moreover, we explicitly describe the Chebyshev centers of a compact subset of an $L_1$-predual space. Various new characterizations of ideals in an $L_1$-predual space are also obtained. In particular, for a compact Hausdorff space $S$ and a subspace $\mathcal {A}$ of $C(S)$ which contains the constant function $1$ and separates the points of $S$, we prove that the state space of $\mathcal {A}$ is a Choquet simplex if and only if $d(\mathcal {A}, {\rm cent}_{C(S)}(F))=0$ for every four-point subset $F$ of $\mathcal {A}$. We also derive characterizations for a compact convex subset of a locally convex topological vector space to be a Choquet simplex.
