---
tag: H1K36H9ZWB
kind: concept
name: Quermassintegral
msc: [52A39, 52A38]
specializes: [QB3Q5FBJ08]
sources:
  - url: https://en.wikipedia.org/wiki/Mixed_volume
---

## Summary

The 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 $B$.
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|F02XA3MYX8]] 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.

## Explanations

### L1 human

The quermassintegrals are a list of $n + 1$ numbers that measure a convex shape in different ways.
In the plane they are the area, half the perimeter, and the constant $\pi$.
In space they are the volume, one third of the surface area, a multiple of the mean width, and $\frac43 \pi$.
For 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.