Arithmetic width counts distinct linear-functional values on lattice points of a convex body; for rational polytopes it grows eventually quasilinearly under dilation, and its computation is claimed to be polynomial or single-exponential in fixed dimension.
On multigraded Hilbert functions and resolutions.Mathematische Proceedings of the Cambridge Philosophical Society, 118:245–257, 1995
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An arithmetic measure of width for convex bodies
Arithmetic width counts distinct linear-functional values on lattice points of a convex body; for rational polytopes it grows eventually quasilinearly under dilation, and its computation is claimed to be polynomial or single-exponential in fixed dimension.