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  "body": "\n## Problem\n\nLet $n \\ge 2$ and let $K, L \\subseteq \\R^n$ be compact [[convex bodies|437FSG3419]] with nonempty interior, both symmetric about the origin.\nAssume that for every hyperplane $H$ through the origin,\n$$\\operatorname{vol}_{n-1}(K \\cap H) \\le \\operatorname{vol}_{n-1}(L \\cap H),$$\nwhere $\\operatorname{vol}_{n-1}$ is the $(n-1)$-dimensional [[Lebesgue measure|2Z2QVWZ262]] on $H$.\nDoes it follow that $\\operatorname{vol}_n(K) \\le \\operatorname{vol}_n(L)$?\n\n## Remarks\n\nThe problem is solved: the answer is yes for $n \\le 4$ and no for $n \\ge 5$.\nFor $n = 2$ the answer is yes: a chord through the center in direction $u$ has length $2\\rho(u)$, where $\\rho$ is the radial function, so the hypothesis gives $K \\subseteq L$.\nLutwak showed that the answer in $\\R^n$ is yes if and only if every origin-symmetric convex body in $\\R^n$ is an [[intersection body|Y4ZE8FVZ8D]].\nCounterexamples in high dimensions were found first by Larman and Rogers, and the remaining dimensions were settled through work of Ball, Bourgain, Giannopoulos, Papadimitrakis, Gardner and Zhang, among others.\nA unified Fourier-analytic solution in all dimensions was given by Gardner, Koldobsky and Schlumprecht.\n\n## Explanations\n\n### L1 human\n\nSuppose that every slice through the center of one symmetric convex body is smaller than the matching slice of another body.\nIt seems obvious that the first body must then have smaller volume.\nSurprisingly, this is true only up to dimension four: in dimension five and higher there are counterexamples.\n",
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      "first_stated": "1956",
      "posed_by": "Herbert Busemann and Clinton Myers Petty"
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      "52A20",
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    "name": "Busemann–Petty problem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Busemann–Petty_problem"
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    "tag": "MV58A1F52E",
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  "msc": [
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  "name": "Busemann–Petty problem",
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  "url": "https://tamnd.github.io/math2/wiki/busemann-petty-problem/",
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