# 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.

... 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.
[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.