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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $K, L$ be nonempty compact convex subsets of $\\R^n$ ([[convex bodies|437FSG3419]]).\nLet $V$ be the [[mixed volume|QB3Q5FBJ08]] and write $V_1(K, L) = V(K, \\dots, K, L)$, with $K$ repeated $n - 1$ times.\nThen\n$$V_1(K, L)^n \\ge \\operatorname{vol}(K)^{n-1} \\operatorname{vol}(L),$$\nwhere $\\operatorname{vol}$ is [[Lebesgue measure|2Z2QVWZ262]].\nIf $K$ and $L$ have nonempty interiors, equality holds if and only if $K$ and $L$ are homothetic.\n\n## Remarks\n\nOne derivation uses the [[Brunn–Minkowski theorem|9B02RPN7XD]].\nIf $\\operatorname{vol}(K) = 0$ the inequality is trivial, so assume $\\operatorname{vol}(K) \u003e 0$.\nThe function $f(t) = \\operatorname{vol}(K + tL)^{1/n}$ is concave on $[0, \\infty)$, so $f'(0) \\ge f(1) - f(0) \\ge \\operatorname{vol}(L)^{1/n}$.\nBy the definition of mixed volumes, $\\operatorname{vol}(K + tL) = \\operatorname{vol}(K) + n V_1(K, L)\\, t + O(t^2)$, so $f'(0) = V_1(K, L) \\operatorname{vol}(K)^{(1-n)/n}$, which gives the inequality.\nIt also follows from the [[Alexandrov–Fenchel inequality|D3AVHVYQW9]].\n\nExample: take $L = B$, the closed unit ball, of volume $\\kappa_n$.\nThen $n V_1(K, B)$ is the surface area $S(K)$, and the inequality becomes $S(K)^n \\ge n^n \\kappa_n \\operatorname{vol}(K)^{n-1}$, the [[isoperimetric inequality|KBX8RBZ626]] for convex bodies.\nFor $n = 2$ this is $\\operatorname{per}(K)^2 \\ge 4 \\pi \\operatorname{area}(K)$.\n\n## Explanations\n\n### L1 human\n\nThe first mixed volume $V_1(K, L)$ measures how fast the volume of $K$ grows when a small multiple of $L$ is added to it.\nMinkowski's first inequality bounds this growth rate from below by the sizes of $K$ and $L$.\nWith $L$ a ball it is the isoperimetric inequality: among convex bodies of given volume, the ball has the least surface area.\n\n### L2 human\n\nThe inequality is a key step in the uniqueness part of the Minkowski problem: two convex bodies with the same surface area measure are translates of each other.\nIt is the case $K_3 = \\cdots = K_n = K$ of a chain of inequalities obtained from the [[Alexandrov–Fenchel inequality|D3AVHVYQW9]].\nMinkowski's second inequality, $V_1(K, L)^2 \\ge \\operatorname{vol}(K)\\, V(K, \\dots, K, L, L)$, is another case of that chain.\n",
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