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  "body": "\n## Definition\n\nLet $K$ be a [[compact set|0R6EWFKRCF]] in $\\C$.\nFor an integer $n \\ge 2$, let\n$$\\delta_n(K) = \\max \\set{ \\prod_{1 \\le i \u003c j \\le n} \\abs{z_i - z_j}^{2/(n(n-1))} : z_1, \\dots, z_n \\in K }.$$\nThe *transfinite diameter* of $K$ is $\\delta(K) = \\lim_{n \\to \\infty} \\delta_n(K)$.\n\n## Remarks\n\nThe maximum exists because $K^n$ is compact and the product is continuous.\nThe sequence $(\\delta_n(K))$ is nonincreasing, so the limit exists.\nTo see this, drop one point from an optimal $(n+1)$-tuple in each of the $n + 1$ possible ways and multiply the resulting $n + 1$ inequalities.\nThe number $\\delta_2(K)$ is the diameter of $K$.\n\nExample: let $K$ be the unit circle and let $\\omega = e^{2 \\pi i / n}$.\nThe square of the Vandermonde determinant of $1, \\omega, \\dots, \\omega^{n-1}$ has absolute value $n^n$, so these points give the product $n^{n/2}$.\nBy Hadamard's inequality no $n$ points of the closed unit disc give more, so $\\delta_n(K) = n^{1/(n-1)}$, which tends to $1$.\n\nNon-example: the limit of $\\delta_n(K)$ is not the diameter in general.\nFor the unit circle $\\delta_2 = 2$, while $\\delta(K) = 1$.\n\nA finite set $K$ has $\\delta(K) = 0$, since for $n$ larger than the number of points two of the $z_i$ coincide.\n",
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