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Given a compact set $K\subset {ℂ}^N$, for each positive integer n, let $V^{\left(n\right)}\left(z\right)$ = $V_K^{\left(n\right)}\left(z\right)$ := sup$1/\left(degp\right)V_{p\left(K\right)}\left(p\left(z\right)\right)$: p holomorphic polynomial, 1 ≤ deg p ≤ n. These “extremal-like” functions $V_K^{\left(n\right)}$ are essentially one-variable in nature and always increase to the “true” several-variable (Siciak) extremal function, $V_K\left(z\right)$:= max[0, sup1/(deg p) log|p(z)|: p holomorphic polynomial, ${\left|\right|p\left|\right|}_K\le 1$]. Our main result is that if K is regular, then all of the functions $V_K^{\left(n\right)}$ are continuous; and their associated Robin functions ${\varrho }_{V_K^{\left(n\right)}}\left(z\right):=limsu{p}_{\left|\lambda \right|\to \infty }[V_K^{\left(n\right)}\left(\lambda z\right)-log\left(\right|\lambda \left|\right)]$ increase to ${\varrho }_K:={\varrho }_{V_K}$ for all z outside a pluripolar set....

We calculate the transfinite diameter for the real unit ball $B_d:=x\in {ℝ}^d:\left|x\right|\le 1$ and the real unit simplex $T_d:=x\in {ℝ}_{+}^d:{\sum }_{j=1}^d{x}_j\le 1.$

We update the state of the subject approximately 20 years after the publication of T. Bloom, L. Bos, C. Christensen, and N. Levenberg, Polynomial interpolation of holomorphic functions in ℂ and ℂⁿ, Rocky Mountain J. Math. 22 (1992), 441-470. This report is mostly a survey, with a sprinkling of assorted new results throughout.