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We prove a characterization of the j-multiplicity of a monomial ideal as thenormalized volume of a polytopal complex. Our result is an extension ofTeissier's volume-theoretic interpretation of the Hilbert-Samuel multiplicityfor m-primary monomial ideals. We also give a description of theepsilon-multiplicity of a monomial ideal in terms of the volume of a region.

The $j$-Multiplicity of Monomial Ideals