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  "body": "\n## Statement\n\nLet $V_1, V_2 \u003e 0$.\nConsider pairs $(E_1, E_2)$ of disjoint [[Caccioppoli sets|W6R7QNEAF1]] in $\\R^3$ with volumes $\\abs{E_1} = V_1$ and $\\abs{E_2} = V_2$.\nFor such a pair, let $A(E_1, E_2) = \\mathcal{H}^2(\\partial^* E_1 \\cup \\partial^* E_2)$ be the total area of the interfaces.\nHere $\\partial^*$ is the reduced boundary and $\\mathcal{H}^2$ is the $2$-dimensional [[Hausdorff measure|Q1TGZK8RN6]], so that a common wall is counted once.\nThen $A$ is least for the standard double bubble with volumes $V_1$ and $V_2$.\nEvery minimizing pair is a standard double bubble, up to a rigid motion and up to sets of measure zero.\n\nThe *standard double bubble* is made of three spherical caps, or of two caps and a flat disc when $V_1 = V_2$.\nThe three pieces have a common boundary circle and meet along it at angles of $120^\\circ$.\n\n## Remarks\n\nMichael Hutchings, Frank Morgan, Manuel Ritoré and Antonio Ros proved the theorem, published in 2002.\nExistence and regularity of a minimizer come from geometric measure theory.\nHutchings showed that a minimizer is a surface of revolution about a line, with strong limits on the number of components of its regions.\nThe authors then excluded the remaining non-standard candidates by an instability argument based on rotating parts of the surface.\nThe case of equal volumes had been proved earlier by Joel Hass and Roger Schlafly with computer help.\nThe analogous result in the plane, with circular arcs in place of spherical caps, had been proved earlier by a group of undergraduates.\nBen Reichardt later extended the theorem to all dimensions.\nFor one region instead of two, the corresponding statement is the [[isoperimetric inequality|KBX8RBZ626]]: the ball encloses a given volume with least area.\n\n## Explanations\n\n### L1 human\n\nBlow two soap bubbles that stick together.\nThey form two spherical pieces separated by a third spherical wall, all meeting along a circle at equal angles.\nThe theorem says that this familiar shape is the cheapest possible way, in surface area, to enclose and separate two given volumes of air.\n\n### L2 human\n\nThe theorem is the simplest case of the problem of least-area clusters, and it shows how hard such problems are.\nThe single-bubble case is classical, but the two-bubble case needed new ideas in the 1990s.\nThe $120^\\circ$ angles agree with Plateau's laws, which every minimizer must satisfy.\nThe analogous problem for three or more bubbles in $\\R^3$ is much harder.\n",
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