  The purpose of this paper is to study the slices of the Okounkov bodies of
Mori dream spaces. First, we analyze all the slices of the Okounkov bodies of
big divisors on Mori dream spaces associated to some admissible flags. As a
byproduct, we obtain their descriptions on Mori dream threefolds. Finally, we
consider its application to the rational polyhedrality of them.
