Affine divisibility of convex sets
Christian Richter ยท Bulletin of the London Mathematical Society ยท 2009
A subset C of a linear topological space X is called m-divisible with respect to a group ๐ข of affine homeomorphisms of X if there exists a disjoint decomposition of C into m subsets Ci pairwise congruent with respect to ๐ข. We ask for the possibility of m-divisibility, mainly for m โ {2, 3, โฆ}, in the following cases: (I) ๐ข contains all affine homeomorphisms, C is bounded, closed, and convex, and m = 2; (II) X is normed and strictly convex, ๐ข consists of all isometries of X, and C is shaped like a ball; and (III) ๐ข is the group of translations and C is closed or open and convex. A geometric characterization of the reflexivity of a Banach space is obtained as a corollary.