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  "body": "\n## Statement\n\nLet $d \\ge 1$ and let $\\mu_1, \\dots, \\mu_d$ be finite Borel measures on $\\R^d$ that are [[absolutely continuous|P0P21KDV6E]] with respect to [[Lebesgue measure|2Z2QVWZ262]].\nThen there is an affine hyperplane $H$ of $\\R^d$ such that, for each $i$, the two open half-spaces bounded by $H$ have measure $\\frac{1}{2} \\mu_i(\\R^d)$ each.\n\n## Remarks\n\nThe usual proof applies the [[Borsuk–Ulam theorem|QHJ0GBQE2E]].\nA unit vector $u = (u_0, u_1, \\dots, u_d)$ of $\\R^{d+1}$ gives the half-space $H_u^+ = \\set{x \\in \\R^d : u_0 + u_1 x_1 + \\cdots + u_d x_d \u003e 0}$.\nThe map $f \\colon S^d \\to \\R^d$ with $f_i(u) = \\mu_i(H_u^+) - \\mu_i(H_{-u}^+)$ is continuous and odd, so it has a zero $u$.\nAt such a zero the hyperplane $u_0 + u_1 x_1 + \\cdots + u_d x_d = 0$ bisects every $\\mu_i$; the points $u = (\\pm 1, 0, \\dots, 0)$ are not zeros as long as the measures are nonzero.\nFor $d = 2$: any two bounded regions of the plane, such as two pancakes, can be cut in half at once by one straight line.\nAbsolute continuity is needed: for $d = 1$ and $\\mu_1$ a single point mass, no point splits $\\mu_1$ into two open halves of measure $\\frac12$ each.\n\n## Explanations\n\n### L1 human\n\nTake a sandwich made of bread, ham and cheese, placed in space in any way.\nOne straight cut with a knife can halve the bread, the ham and the cheese at the same time.\nIn the plane, two shapes can always be halved by one straight line.\nIn general, $d$ objects in $d$-dimensional space can be halved by one flat cut.\n\n### L2 human\n\nThe theorem is a standard first application of the [[Borsuk–Ulam theorem|QHJ0GBQE2E]] to combinatorial geometry.\nIt has a discrete version for finite point sets, where the hyperplane may contain some points and each open side has at most half of each set.\nIt is used for partitioning point sets in computational geometry, and it is related to the [[necklace splitting problem|WFS3PHFHA1]].\n",
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