Continuity of a certain invariant of a measure on a CAT(0) space

Abstract

For a finitely supported probability measure $\mu$ on a complete CAT(0) space $Y$ , Izeki and Nayatani defined an invariant $\delta(\mu) \in [0,1]$ in [1]. The supremum of those for all such measures on $Y$ is an invariant of $Y$ , called the Izeki-Nayatani invariant, which plays an important role in the study of fixed-point property of groups. In this paper, we establish continuity of $\delta$ on the space of finitely supported probability measures. We prove the lower-semicontinuity of $\delta$ with respect to the $(L^2-)$ Wasserstein metric, and continuity with respect to some metric which induces a stronger topology.