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  "body": "\n## Definition\n\nLet $K$ be a [[compact set|0R6EWFKRCF]] in $\\C$ with at least $n \\ge 2$ points.\nAn $n$-tuple $(z_1, \\dots, z_n)$ of points of $K$ is a system of *Fekete points* of $K$ if\n$$\\prod_{1 \\le i \u003c j \\le n} \\abs{z_i - z_j} = \\max \\set{ \\prod_{1 \\le i \u003c j \\le n} \\abs{w_i - w_j} : w_1, \\dots, w_n \\in K }.$$\n\n## Remarks\n\nA system of Fekete points exists for each $n$, since $K^n$ is compact and the product is continuous.\nIt need not be unique: a rotation of Fekete points of a circle centred at $0$ gives Fekete points again.\nThe points are distinct, because the maximum is positive.\nThe maximum equals $\\delta_n(K)^{n(n-1)/2}$, where $\\delta_n(K)$ is the $n$-th diameter in the definition of the [[transfinite diameter|VSGTSQ7RY8]].\n\nExample: on the unit circle, the $n$-th roots of unity $1, \\omega, \\dots, \\omega^{n-1}$ with $\\omega = e^{2\\pi i/n}$ are Fekete points.\nThe product of their mutual distances is the absolute value of their Vandermonde determinant, which equals $n^{n/2}$.\nHadamard's inequality bounds the Vandermonde determinant of any $n$ points of the closed unit disc by $n^{n/2}$, so the maximum is attained.\n\nNon-example: for $n = 2$ on the unit circle, the points $1$ and $i$ are not Fekete points, since $\\abs{1 - i} = \\sqrt 2 \u003c 2 = \\abs{1 - (-1)}$.\n\nMaximizing the product is the same as minimizing the discrete logarithmic energy $\\sum_{i \u003c j} \\log(1/\\abs{z_i - z_j})$.\n",
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