# Origin of the term quermassintegral'.

What is the origin of the term quermassintegral'? I think this is a german word. What would be its literal translation in English?

The definition of quermassintegrals from wikipedia:

Let $K\subset\mathbb{R}^n$ be a convex body, and let $B\subset\mathbb{R}^n$ be the unit ball. The mixed volume $$W_j(K) = V(\overset{n-j \text{ times}}{\overbrace{K,K, \ldots,K}}, \overset{j\text{ times}}{\overbrace{B,B,\ldots,B}})$$ is called the $j$-th quermassintegral of K.

## 2 Answers

Gruber writes, in ch.6 of his Convex and Discrete Geometry (Springer, 2007):

... the general formulae of Kubota ... express the quermassintegrals $W_i(C)$ of $C$ as the mean of the $(d-i)$-dimensional volumes of the projections of $C$ onto $(d-i)$-dimensional linear subspaces. These volumes are called Quermaße in German.

Quer means crosswise, or transverse; maße means mass.

Also, Schneider writes, in §4.2 of his Convex Bodies: The Brunn–Minkowski Theory (Cambridge UP, 1993):

[The term quermassintegrals] comes from the German 'Quermaß', which can be the measure of either a cross-section or a projection. The reason for this terminology will be clear when certain integral-geometric interpretations of the functions $W_i$ have been obtained in Section 4.5.

It is definitely a German term. In general, quer- is used with the meaning across. My technical dictionary translates Quermass as cross-sectional measure, and Quermassintegral as mean cross-sectional measure. It also attributes the term to Minkowski.