For a finite dimensional algebra A over a field k, the 2-term silting
complexes of A gives a simplicial complex Δ(A) called the
g-simplicial complex. We give tilting theoretic interpretations of the
h-vectors and Dehn-Sommerville equations of Δ(A). Using g-vectors of
2-term silting complexes, Δ(A) gives a nonsingular fan Σ(A) in
the real Grothendieck group K0​(projA)R​ called the
g-fan. For example, the fan of g-vectors of a cluster algebra is given by
the g-fan of a Jacobian algebra of a non-degenerate quiver with potential. We
give several properties of Σ(A) including idempotent reductions,
sign-coherence, Jasso reductions and a connection with Newton polytopes of
A-modules. Moreover, Σ(A) gives a (possibly infinite and non-convex)
polytope P(A) in K0​(projA)R​ called the g-polytope
of A. We call Ag-convex if P(A) is convex. In this case, we show that
it is a reflexive polytope, and that the dual polytope is given by the 2-term
simple minded collections of A.
We give an explicit classification of g-convex algebras of rank 2. We
classify algebras whose g-polytopes are smooth Fano. We classify classical
and generalized preprojective algebras which are g-convex, and also describe
their g-polytope as the dual polytopes of short root polytopes of type A
and B. We also classify Brauer graph algebras which are g-convex, and
describe their g-polytopes as root polytopes of type A and C.Comment: 70 page