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Alexander’s projective capacity for polydisks and ellipsoids in $ℂ^{N}$

Mieczysław Jędrzejowski — 1995

Annales Polonici Mathematici

Alexander’s projective capacity for the polydisk and the ellipsoid in $ℂ^{N}$ is computed. Sharper versions of two inequalities concerning this capacity and some other capacities in $ℂ^{N}$ are given. A sequence of orthogonal polynomials with respect to an appropriately defined measure supported on a compact subset K in $ℂ^{N}$ is proved to have an asymptotic behaviour in $ℂ^{N}$ similar to that of the Siciak homogeneous extremal function associated with K.

The homogeneous transfinite diameter of a compact subset of $ℂ^{N}$

Mieczysław Jędrzejowski — 1991

Annales Polonici Mathematici

Let K be a compact subset of $ℂ^{N}$ . A sequence of nonnegative numbers defined by means of extremal points of K with respect to homogeneous polynomials is proved to be convergent. Its limit is called the homogeneous transfinite diameter of K. A few properties of this diameter are given and its value for some compact subsets of $ℂ^{N}$ is computed.