Pluripotential theory and convex bodies: a Siciak-Zaharjuta theorem

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Official URL: http://dx.doi.org/10.1007/s40315-020-00345-6

We work in the setting of weighted pluripotential theory arising from polynomials associated to a convex body P in (R+)d. We define the logarithmic indicator function on Cd: HP(z):=supJ∈Plog|zJ|:=supJ∈Plog[|z1|j1⋯|zd|jd]and an associated class of plurisubharmonic (psh) functions: LP:={u∈PSH(Cd):u(z)-HP(z)=O(1),|z|→∞}.We first show that LP is not closed under standard smoothing operations. However, utilizing a continuous regularization due to Ferrier which preserves LP, we prove a general Siciak–Zaharjuta type-result in our P-setting: the weighted P-extremal function VP,K,Q(z):=sup{u(z):u∈LP,u≤QonK}associated to a compact set K and an admissible weight Q on K can be obtained using the subclass of LP arising from functions of the form 1degP(p)log|p|.