  Based on the work of Okounkov, Kaveh-Khovanskii and Lazarsfeld-Mustata
independently associated a convex body, called the Okounkov body, to a big
divisor on a normal projective variety with respect to an admissible flag.
Although the Okounkov bodies carry rich positivity data of big divisors, they
only provide information near a single point. The purpose of this paper is to
introduce a convex body of a big divisor that is effective in handling the
positivity theory associated with multi-point settings. These convex bodies
open the door to approach the local positivity theory at multiple points from a
convex-geometric perspective. We study their properties and shapes, and
describe local positivity data via them. Finally, we observe the irrationality
of Seshadri constants with the help of a relation between Nakayama constants
and Seshadri constants.
