2
$\begingroup$

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.

$\endgroup$
3
$\begingroup$

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.

$\endgroup$
2
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.