  In recent years, the study of Newton-Okounkov bodies on normal varieties has
become a central subject in the asymptotic theory of linear series, after its
introduction by Lazarsfeld-Mustata and Kaveh-Khovanskii. One reason for this is
that they encode all numerical equivalence information of divisor classes (by
work of Jow). At the same time, they can be seen as local positivity
invariants, and K\"uronya-Lozovanu have studied them in depth from this point
of view. We determine what information is encoded by the set of all
Newton-Okounkov bodies of a big divisor with respect to flags centered at a
fixed point of a surface, by showing that it determines and is determined by
the numerical equivalence class of the divisor up to negative components in the
Zariski decomposition that do not go through the fixed point.
