Affine invariant points

Abstract

We answer in the negative a question by Grünbaum who asked if there exists a finite basis of affine invariant points. We give a positive answer to another question by Grünbaum about the "size" of the set of all affine invariant points. Related, we show that the set of all convex bodies K, for which the set of affine invariant points is all of R n , is dense in the set of convex bodies. Crucial to establish these results are new affine invariant points, not previously considered in the literature.

Halfspace depth and floating body

“…The following theorems answer Grünbaum's questions (i) and (ii). They can be found in Meyer et al [123].…”

“…Moreover, in Meyer et al [123,Theorem 2] it was shown that for convex bodies K with dim(P d (K)) = d − 1 a positive answer to Grünbaum's question (iii) above holds, i.e. F d (K) = P d (K).…”

Dual Affine invariant points

“…In two preceding papers, [12] and [13], we answered some of Grünbaum's questions: The dimension of the space of affine invariant points is infinte and there are convex bodies K in R n such that every point in R n is an affine invariant point of K. More importantly, we showed in some cases that the presence of many affine invariant points means that the convex body lacks symmetry.…”

“…At the same time, these are fundamental invariants of convex sets. They are, for instance, useful to characterize properties of symmetry or of non symmetry of convex bodies (e.g., [12] and [13]). The more different affine invariant points a convex body has the less symmetric it is.…”

“…More details on affine invariant points, and some results that we shall use here, can be found in [13].…”