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  "body": "\n## Statement\n\nLet $B \\subseteq \\R^n$ be a nonempty [[connected|0SPE8WNWFV]] [[open set|02BH3MCBQB]], and let $\\operatorname{ch}(B)$ be its [[convex hull|0FQSDC2EQ2]].\nFor a set $S \\subseteq \\R^n$ let $T_S = \\set{x + iy \\in \\C^n : x \\in S, \\ y \\in \\R^n}$ be the tube over $S$.\nThen every [[holomorphic function|507C34W0ZT]] on $T_B$ is the restriction of a unique holomorphic function on $T_{\\operatorname{ch}(B)}$.\n\n## Remarks\n\nUniqueness holds because $T_{\\operatorname{ch}(B)}$ is connected and contains the nonempty open set $T_B$.\nClassical proofs push the function across the convex hull with Cauchy integrals over families of analytic discs, in the spirit of the continuity principle.\nAnother route uses that a tube which is a domain of holomorphy has a convex base.\nOn such a tube, $-\\log$ of the distance to the boundary is plurisubharmonic.\nIt does not depend on $y$, so it is a convex function of $x$.\n\nExample: let $B = \\set{x \\in \\R^2 : 1 \u003c \\abs{x} \u003c 2}$, an annulus.\nIts convex hull is the disc $\\abs{x} \u003c 2$, so every holomorphic function on $T_B$ extends to the tube over the disc.\n\nConsequently, a tube $T_B$ with connected open base is a domain of holomorphy exactly when $B$ is convex.\nFor $n = 1$ the theorem is empty, since connected open subsets of $\\R$ are intervals and already convex.\n\n## Explanations\n\n### L1 human\n\nA tube domain is determined by its real part: the imaginary part may be anything.\nThe theorem says that holomorphic functions on a tube automatically extend to the tube over the convex hull of the base.\nSo, for tubes, being a natural domain of holomorphic functions is the same as ordinary convexity of the base.\n\n### L2 human\n\nTube domains appear in Fourier and Laplace analysis: the Laplace transform $z \\mapsto \\int e^{z_1 t_1 + \\cdots + z_n t_n} \\, d\\mu(t)$ of a measure $\\mu$ on $\\R^n$ is holomorphic on a tube whose base is convex.\nThe theorem explains why convex bases are the natural ones.\nIt is a basic example of the computation of an envelope of holomorphy: the envelope of $T_B$ is $T_{\\operatorname{ch}(B)}$.\n",
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