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  "body": "\n## Summary\n\nThe quermassintegrals $W_0(K), \\dots, W_n(K)$ of a convex body $K$ in $\\R^n$ are [[mixed volumes|QB3Q5FBJ08]] of $K$ and the unit ball $B$.\nIn $W_j(K)$, the body $K$ is taken $n - j$ times and $B$ is taken $j$ times.\nThey are the coefficients of the [[Steiner formula|F02XA3MYX8]] for the volume of the parallel body $K + tB$.\nThey include the volume, a multiple of the surface area and a multiple of the mean width.\nEach one is an average of volumes of projections of $K$ onto subspaces.\n\n## Explanations\n\n### L1 human\n\nThe quermassintegrals are a list of $n + 1$ numbers that measure a convex shape in different ways.\nIn the plane they are the area, half the perimeter, and the constant $\\pi$.\nIn space they are the volume, one third of the surface area, a multiple of the mean width, and $\\frac43 \\pi$.\nFor a ball of radius $r$ in $\\R^n$, the $j$-th quermassintegral is $\\kappa_n r^{n-j}$, where $\\kappa_n$ is the volume of the unit ball.\n",
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