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  "body": "\n## Statement\n\nLet $n \\ge 2$ and let $K_1, K_2, \\dots, K_n$ be nonempty compact convex subsets of $\\R^n$ ([[convex bodies|437FSG3419]]).\nLet $V$ denote the [[mixed volume|QB3Q5FBJ08]].\nThen\n$$V(K_1, K_2, K_3, \\dots, K_n)^2 \\ge V(K_1, K_1, K_3, \\dots, K_n)\\, V(K_2, K_2, K_3, \\dots, K_n).$$\n\n## Remarks\n\nThe inequality was proved by Alexandrov, and independently by Fenchel, in the 1930s.\nAlexandrov's proofs treat polytopes or smooth bodies first, by induction on the dimension, and then pass to general bodies by approximation.\nKhovanskii and Teissier found a proof through the Hodge index theorem applied to toric varieties.\n\nExample: for $n = 2$ the inequality reads $V(K, L)^2 \\ge \\operatorname{area}(K) \\operatorname{area}(L)$.\nWith $L$ the unit disk, $V(K, L) = \\frac12 \\operatorname{per}(K)$, and one gets $\\operatorname{per}(K)^2 \\ge 4 \\pi \\operatorname{area}(K)$, the planar [[isoperimetric inequality|KBX8RBZ626]].\n\n## Explanations\n\n### L1 human\n\nMixed volumes measure how several convex shapes interact.\nThe Alexandrov–Fenchel inequality says that a mixed volume of two different shapes is at least the geometric mean of the \"pure\" mixed volumes of each shape with itself.\nIt is a far-reaching relative of the Cauchy–Schwarz inequality and of the isoperimetric inequality.\n\n### L2 human\n\nApplied with the remaining arguments chosen among $K_1$ and $K_2$, it shows that the sequence $V(K_1[i], K_2[n - i])$, for $i = 0, \\dots, n$, is log-concave in $i$.\nIt implies [[Minkowski's first inequality|9M8XJX4DQ7]] and the [[Brunn–Minkowski theorem|9B02RPN7XD]] for convex bodies.\nStanley used it to prove that the numbers of linear extensions of a poset with a fixed element at a given position form a log-concave sequence.\nIts equality cases are understood only in special situations, and their full description is a hard problem.\n",
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