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  "body": "\n## Summary\n\nLet $K_1, \\dots, K_r$ be nonempty compact convex sets of $\\R^n$ and $\\lambda_i \\ge 0$.\nThe volume of the [[Minkowski sum|SXS6D87PHE]] $\\lambda_1 K_1 + \\cdots + \\lambda_r K_r$ is a homogeneous polynomial of degree $n$ in the $\\lambda_i$.\nIts coefficients are the mixed volumes $V(K_{i_1}, \\dots, K_{i_n})$.\nMixed volumes unify volume, surface area, mean width and many other measurements of convex bodies.\nThey satisfy deep inequalities, above all the [[Alexandrov–Fenchel inequality|D3AVHVYQW9]].\n\n## Explanations\n\n### L1 human\n\nBlow up two convex shapes by factors $\\lambda$ and $\\mu$ and add them.\nThe area of the result is $A(K) \\lambda^2 + 2 V(K, L) \\lambda \\mu + A(L) \\mu^2$, a quadratic expression.\nThe middle coefficient $V(K, L)$ is the mixed area of $K$ and $L$: it measures how the two shapes interact.\nWhen $L$ is the unit disk, $V(K, L)$ is half the perimeter of $K$.\nSo perimeter, and in higher dimensions surface area, are special mixed volumes.\n",
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