The problem of deciding if a given triangulation of a sphere is realizable as the boundary sphere of a simplicial, convex polytope is known as the “Simplicial Steinitz problem”. This is an example of a problem of geometric combinatorics which links together areas of mathematics as distant as toric topology, combinatorial optimization, convex polytopes, algebraic geometry, topological combinatorics, discrete and computational geometry, etc.
It is known (by indirect and non-constructive arguments) that a vast majority of triangulated spheres are “non-polytopal”, in the sense that they are not combinatorially isomorphic to the boundary of a convex polytope. This holds, in particular, for Bier spheres Bier(K) (named after Thomas Bier), the (n-2)-dimensional, combinatorial spheres on 2n-vertices, constructed with the aid of simplicial complexes K on n vertices.
Emphasizing connections with polyhedral products and toric topology, we review „hidden geometry” of Bier spheres by describing their natural geometric realizations, compute their volume, describe an effective criterion for their “strong polytopality”, and associate to Bier(K) a natural coarsening Fan(K) of the Braid fan. We also establish a connection of Bier spheres of maximal volume with recent generalizations of the classical Van Kampen-Flores theorem and clarify the role of Bier spheres in the theory of generalized permutohedra.