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  "body": "\n## Summary\n\nA logarithmically convex domain is a [[Reinhardt domain|WA0SE56FVH]] whose image under the map $z \\mapsto (\\log\\abs{z_1}, \\dots, \\log\\abs{z_n})$ is a convex subset of $\\R^n$.\nThe [[domain of convergence of a power series|D8S8JWE45R]] always has this property.\nFor complete Reinhardt domains, logarithmic convexity is exactly the condition that the domain is a domain of holomorphy.\nThe notion is close to that of a [[logarithmically convex function|8FRWATNNMX]]: a function is logarithmically convex when its logarithm is convex.\n\n## Explanations\n\n### L1 human\n\nA Reinhardt domain can be drawn by the absolute values of its points.\nTake logarithms of these absolute values: the picture then lives in $\\R^n$.\nThe domain is logarithmically convex when this new picture is convex, that is, contains the segment between any two of its points.\nFor example, $\\set{\\abs{z_1 z_2} \u003c 1}$ becomes the half-plane $\\set{x_1 + x_2 \u003c 0}$, which is convex.\nA union of two thin polydiscs crossing like a plus sign becomes an L-shape, which is not convex.\n",
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