Sums of congruent convex bodies

  • Date: 11/02/2007

Rolf Schneider (University of Freiburg)


University of British Columbia


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.)

Other Information: 

PIMS Distinguished Lecture 2007