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Article Dans Une Revue Mathematica Scandinavica Année : 2021

On transfinite diameters in $\mathbb{C}^{d}$ for generalized notions of degree

Résumé

We give a general formula for the $C$-transfinite diameter $\delta_C(K)$ of a compact set $K\subset \mathbb{C}^2$ which is a product of univariate compacta where $C\subset (\mathbb{R}^+)^2$ is a convex body. Along the way we prove a Rumely type formula relating $\delta_C(K)$ and the $C$-Robin function $\rho_{V_{C,K}}$ of the $C$-extremal plurisubharmonic function $V_{C,K}$ for $C \subset (\mathbb{R}^+)^2$ a triangle $T_{a,b}$ with vertices $(0,0)$, $(b,0)$, $(0,a)$. Finally, we show how the definition of $\delta_C(K)$ can be extended to include many nonconvex bodies $C\subset \mathbb{R}^d$ for $d$-circled sets $K\subset \mathbb{C}^d$, and we prove an integral formula for $\delta_C(K)$ which we use to compute a formula for $\delta_C(\mathbb{B})$ where $\mathbb{B}$ is the Euclidean unit ball in $\mathbb{C}^2$.

Dates et versions

hal-03523818 , version 1 (04-04-2023)

Identifiants

Citer

Norman Levenberg, Franck Wielonsky. On transfinite diameters in $\mathbb{C}^{d}$ for generalized notions of degree. Mathematica Scandinavica, 2021, 127 (2), pp.337-360. ⟨10.7146/math.scand.a-126053⟩. ⟨hal-03523818⟩
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