## Sums of congruent convex bodies

- Date: 10/26/2007

Rolf Schneider (University of Freiburg)

University of Alberta

The Minkowski linear combination is a fundamental operation for convex

bodies. Further basic structures on the space of convex bodies are the

topology induced by the Hausdorff metric, and the operation of the

group of rigid motions. Suppose we have only one convex body B at our

hands and want to produce other convex bodies from it by using just

these basic operations, that is, taking Minkowski linear combinations

of congruent copies of B, and limits. How far do we get? Not very far:

we obtain only a nowhere dense class of convex bodies (example: if B is

a segment, we obtain the zonoids). What if we allow also subtractions',

in the following sense: we say that B generates the convex body K if K

can be represented by K + M_1 = M_2 , where M_1 and M_2 are limits of

Minkowski linear combinations of congruent copies of B. Our aim is to

characterize the convex bodies B that generate a dense class of convex

bodies, and to show that they are also dense. This involves some

elementary harmonic analysis. (Joint work with Franz Schuster.)

PIMS Distinguished Lecture 2007