Constructing infinite-dimensional polytopes using Gale transforms (Q1808762)

The infinite dimensional analogue of the Gale transform may be defined in a natural way for compact convex subsets of a Banach space with countably many extreme points such that the linear space of affine dependencies is finite-dimensional (so-called \(\alpha \)-polytopes). This paper describes the construction of an \(\alpha \)-polytope with a given Gale transform. This process depends critically on the construction of a ``Gale basis''. Since the Gale transform contains only algebraic information about the polytope, a single transform can give rise to topologically distinct polytopes. This is illustrated by a concrete example.