  For a finite dimensional algebra $A$ over a field $k$, the 2-term silting
complexes of $A$ gives a simplicial complex $\Delta(A)$ called the
$g$-simplicial complex. We give tilting theoretic interpretations of the
$h$-vectors and Dehn-Sommerville equations of $\Delta(A)$. Using $g$-vectors of
2-term silting complexes, $\Delta(A)$ gives a nonsingular fan $\Sigma(A)$ in
the real Grothendieck group $K_0(\mathsf{proj} A)_{\mathbb{R}}$ called the
$g$-fan. We give several basic properties of $\Sigma(A)$ including
sign-coherence, sign decomposition, idempotent reductions, Jasso reductions,
pairwise positivity and a connection with Newton polytopes of $A$-modules.
Moreover, $\Sigma(A)$ gives a (possibly infinite and non-convex) polytope
$P(A)$ in $K_0(\mathsf{proj} A)_{\mathbb{R}}$ called the $g$-polytope of $A$.
We call $A$ $g$-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 $A$. There are precisely 7 convex $g$-polyogons up to
isomorphism. We give a classification of algebras whose $g$-polytopes are
smooth Fano.
  We study $g$-fans and $g$-polytopes of two important classes of algebras. We
show that the $g$-fan of a classical or generalized preprojective algebra is
given by the Coxeter fan. It is $g$-convex if and only if it is of type $A$ or
$B$, and in this case, its $g$-polytope is the dual polytope of the short root
polytope. Moreover we classify Brauer graph algebras which are $g$-convex, and
describe their $g$-polytopes as the root polytopes of type $A$ or $C$.
