  Starting from the data of a big line bundle $L$ on a projective manifold $X$
with a choice of $N\geq 1$ different points on $X$ we provide a new
construction of $N$ Okounkov bodies which encodes important geometric features
of ($L\to X,p_{1},\dots,p_{N}$) such as the volume of $L$, the (moving)
multipoint Seshadri constant of $L$ at $p_{1},\dots,p_{N}$, and the possibility
to construct K\"ahler packings centered at $p_{1},\dots,p_{N}$. Toric manifolds
and surfaces are examined in detail.
