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In tropical geometry, is there a notion of volume. Maybe one with some of the properties as found in classical convex geometry? If so, is there a good reference that elaborates on this question.

I found this question here, but apparently the answers did not consider volume.

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I think a certain generalization of volume in tropical geometry is given by Monge-Ampere measure. See http://www2.math.su.se/reports/2000/10/2000-10.pdf (M. Passare and H. Rullgard, Amoebas, Monge-Ampere measures, and triangulations of the Newton polytope) and http://arxiv.org/abs/1008.2856 (A. Lagerberg, Super currents and tropical geometry).

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