Genocchi numbers and f-vectors of simplicial balls

The aim of this note is to investigate f -vectors of simplicial balls, especially the relations between interior and boundary faces. For a simplicial ball B we denote by fi(B) the number of i-dimensional faces. The boundary ∂B of B is a simplicial sphere with face numbers fi(∂B). We also define fi(int B) := fi(B) − fi(∂B) although the interior int B of B is not a polyhedral complex. For simplicial spheres the Dehn-Sommerville relations give non-trivial dependences for the f -vector. McMullen and Walkup gave in [13] a similar equation for simplicial balls which relates the f -vector of the boundary ∂B with the f -vector of B (see also Billera and Björner [3] and Klain [10]). The most important examples of simplicial balls are triangulated polytopes. Special interest on connections between the f -vector of the boundary and the interior occurs for example when the polytope is simplicial and so the f -vector of the boundary is fixed (see McMullen [12]), but also in other cases, e.g. for triangulations without interior edges of low dimension [8]. The f -vector of the interior of a simplicial ball occurs in the study of the combinatorics of certain unbounded polyhedra [8, Section 2]: It is dual to the f -vector of the polyhedral complex of their bounded faces. A very important special case is the tight span of a finite metric space [5, 8]. We will give a relation between the f -vector of the boundary and the interior of a simplicial ball directly in terms of the f -vector. The most interesting point about our equation is the occurrence of the Genocchi numbers G2n: