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  "body": "\n## Summary\n\nBrunn–Minkowski theory is the part of convex geometry that studies [[convex bodies|437FSG3419]] through the interplay of volume and [[Minkowski addition|SXS6D87PHE]].\nIts central objects are the [[mixed volumes|QB3Q5FBJ08]], the coefficients of the polynomial that gives the volume of a Minkowski combination of convex bodies.\nIts central results are inequalities between them, starting from the [[Brunn–Minkowski theorem|9B02RPN7XD]] and leading to the Alexandrov–Fenchel inequality.\nSurface area, mean width, quermassintegrals and the isoperimetric inequality all fit into this framework.\n\n## Explanations\n\n### L1 human\n\nAdd two convex shapes by adding each point of one to each point of the other, and ask how the volume of the result depends on the pieces.\nThe volume of $s K + t L$ turns out to be a polynomial in $s$ and $t$, and its coefficients measure how $K$ and $L$ interact.\nFor a convex body $K$ and the unit ball $B$, the volume of $K + t B$ is a polynomial in $t$ whose coefficients include the volume and the surface area of $K$.\nBrunn–Minkowski theory studies these coefficients and the inequalities between them.\n",
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