The notion of metric measure boundary for a metric measure space was introduced by Kapovitch, Lytchak and Petrunin as an averaged deviation of the volume of small balls with respect to the corresponding Euclidean ones. Among their motivations there was the problem of the existence of infinite geodesics on Alexandrov spaces, raised by Perelman and Petrunin in 1996. In particular, Kapovitch, Lytchak and Petrunin showed that ``many’’ infinite geodesics exist on any Alexandrov space, provided its metric measure boundary vanishes. They also conjectured that this should always be the case whenever the (topological) boundary is empty and established the result in some particular cases.

In this talk I will discuss joint work with E. Bruè and A. Mondino where we proved that the metric measure boundary vanishes, more in general, for any non collapsed RCD space with empty boundary.