Transfinite diameter, Chebyshev constant and energy on locally compact spaces

2007/04/06 by Bálint Farkas, Balint Farkas, Farkas, Balint +3
Mathematics · Physics and Astronomy · #28A12 #31C15 #54D45 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Quantum Mechanics and Applications #Spectral Theory in Mathematical Physics #math.CA #msc:28A12 #msc:31C15 #msc:54D45

paper · pdf · doi:10.48550/arxiv.0704.0859

arxiv created 2007/04/06 · openalex publication_date 2007/04/06 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the relationship between transfinite diameter, Chebyshev constant and Wiener energy in the abstract linear potential analytic setting pioneered by Choquet, Fuglede and Ohtsuka. It turns out that, whenever the potential theoretic kernel has the maximum principle, then all these quantities are equal for all compact sets. For continuous kernels even the converse statement is true: if the Chebyshev constant of any compact set coincides with its transfinite diameter, the kernel must satisfy the maximum principle. An abundance of examples is provided to show the sharpness of the results.

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