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Towards the boundary of the big cone, Newton-Okounkov bodies do not varycontinuously and in fact the body of a boundary class is not well defined.Using the global Okounkov body one can nonetheless define a numericalinvariant, the numerical Newton-Okounkov body. We show that if a normalprojective variety has a rational polyhedral global Okounkov body, it admits aMinkowski basis provided one includes numerical Newton-Okounkov bodies abovenon-big classes. Under the same assumption, we also show that the dimension ofthe numerical Newton-Okounkov body is the numerical Kodaira dimension.

On numerical Newton-Okounkov bodies and the existence of Minkowski bases