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  "body": "\n## Definition\n\nLet $D \\subseteq \\C^n$ be a [[Reinhardt domain|WA0SE56FVH]] with centre $0$.\nLet $D^* = \\set{z \\in D : z_j \\ne 0 \\text{ for } j = 1, \\dots, n}$, and let $\\lambda \\colon D^* \\to \\R^n$ be the map $\\lambda(z) = (\\log\\abs{z_1}, \\dots, \\log\\abs{z_n})$.\nThe domain $D$ is *logarithmically convex* if $\\lambda(D^*)$ is a [[convex set|02AS45T3H9]] in $\\R^n$.\n\n## Remarks\n\nExample: for the unit ball $\\mathbb{B}^2$, $\\lambda(\\mathbb{B}^{2*}) = \\set{x \\in \\R^2 : e^{2x_1} + e^{2x_2} \u003c 1}$, a sublevel set of the convex function $e^{2x_1} + e^{2x_2}$, hence convex.\nEvery polydisc with centre $0$ is logarithmically convex, since its image is a product of half-lines $(-\\infty, \\log r_j)$.\nNon-example: let $D = \\set{\\abs{z_1} \u003c 1, \\abs{z_2} \u003c e^{-2}} \\cup \\set{\\abs{z_1} \u003c e^{-2}, \\abs{z_2} \u003c 1}$, a Reinhardt domain.\nThe points $(e^{-1}, e^{-3})$ and $(e^{-3}, e^{-1})$ lie in $D$, with images $(-1, -3)$ and $(-3, -1)$.\nThe midpoint $(-2, -2)$ of these images is the image of $(e^{-2}, e^{-2})$, which is not in $D$, so $\\lambda(D^*)$ is not convex.\nFor a complete Reinhardt domain, logarithmic convexity is equivalent to being a domain of holomorphy.\n",
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