Quermassintegral

Summary

The quermassintegrals $W_0(K), \dots, W_n(K)$ of a convex body $K$ in $\R^n$ are mixed volumes of $K$ and the unit ball $B$. In $W_j(K)$, the body $K$ is taken $n - j$ times and $B$ is taken $j$ times. They are the coefficients of the Steiner formula for the volume of the parallel body $K + tB$. They include the volume, a multiple of the surface area and a multiple of the mean width. Each one is an average of volumes of projections of $K$ onto subspaces.