2020/03/25 by N. Levenberg, Levenberg, N., F. Wielonsky +1
Mathematics · #Analytic and geometric function theory #Ball (mathematics) #Combinatorics #Compact space #Complex Variables (math.CV) #Convex body #Convex hull #Discrete mathematics #Euclidean geometry #FOS: Mathematics #Geometry #Geometry and complex manifolds #Holomorphic and Operator Theory #Mathematical analysis #Mathematics #Regular polygon #Transfinite number #Type (biology) #Unit sphere #math.CV
paper · pdf · doi:10.48550/arxiv.2003.11607
18 pages, 1 figure
arxiv created 2020/03/25 · openalex publication_date 2020/03/25 · arxiv updated 2020/03/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We give a general formula for the C-transfinite diameter δC(K) of a compact set K⊂ ℂ2 which is a product of univariate compacta where C⊂ (ℝ+)2 is a convex body. Along the way we prove a Rumely type formula relating δC(K) and the C-Robin function ρ_VC,K of the C-extremal plurisubharmonic function VC,K for C ⊂ (ℝ+)2 a triangle Ta,b with vertices (0,0), (b,0), (0,a). Finally, we show how the definition of δC(K) can be extended to include many nonconvex bodies C⊂ ℝd for d-circled sets K⊂ ℂd, and we prove an integral formula for δC(K) which we use to compute a formula for the C-transfinite diameter of the Euclidean unit ball \mathbbB⊂ ℂ2.