For graded families of monomial or invariant ideals, the asymptotic resurgence is the dilation threshold at which one associated convex body fails to sit inside the other.
Villarreal, A duality theorem for the ic-resurgence of edge ideals
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Resurgence number and convex body associated to pairs of graded families of ideals
For graded families of monomial or invariant ideals, the asymptotic resurgence is the dilation threshold at which one associated convex body fails to sit inside the other.