Intrinsic volume

Summary

The intrinsic volumes $V_0(K), \dots, V_n(K)$ of a convex body $K$ in $\R^n$ are rescaled quermassintegrals. The scaling makes them independent of the dimension of the ambient space. $V_n$ 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. By Hadwiger's theorem they span all continuous valuations on convex bodies that are invariant under rigid motions.