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  "body": "\n## Summary\n\nThe intrinsic volumes $V_0(K), \\dots, V_n(K)$ of a convex body $K$ in $\\R^n$ are rescaled [[quermassintegrals|H1K36H9ZWB]].\nThe scaling makes them independent of the dimension of the ambient space.\n$V_n$ is the volume, $2 V_{n-1}$ is the surface area, and $V_0$ is the Euler characteristic, equal to $1$ for every nonempty convex body.\nBy [[Hadwiger's theorem|7V2X68AMEY]] they span all continuous valuations on convex bodies that are invariant under rigid motions.\n\n## Explanations\n\n### L1 human\n\nIntrinsic volumes are the natural sizes of a convex shape in each dimension.\nFor a box with sides $a, b, c$ they are $abc$ (volume), $ab + bc + ca$ (half the surface area), $a + b + c$ (a total length), and $1$.\nIf the box is flattened into a rectangle with sides $a, b$, one gets $0$, $ab$, $a + b$ and $1$, exactly the intrinsic volumes of the rectangle in the plane.\nThis independence of the surrounding space is what \"intrinsic\" means.\n",
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