it is known that tropicalization of a variety(irreducible and subvariety of some torus.) is a support of a polyhedral complex. I wonder which kinds of polyhedra can occur in this polyhedral complex. In other words, if P is any polyhedron of dimension d, then does some variety X always exist such that P is in a polyhedral complex of Trop(X)?



Polyhedra from a tropical variety
Provided that the facets of $P$ have integral normal vectors, this is always the case.
To construct such a tropical variety, first let $\Sigma$ be the normal fan to $P$.
This comes along with a piecewise linear function $\psi$ defined by 
$\psi(n)=-inf\{ \langle n, m \rangle\,|\, m\in P\}$. Now choose any lattice polytope
$Q$ containing the origin as an interior point, and restrict $\psi$ to $Q$.
Take the tropical polynomial $$f=\sum\_{m\in Q\cap {\bf Z}^n} \psi(m) z^m.$$
It is then a simple exercise in convex geometry to see that this tropical polynomial
defines a tropical variety, which, when viewed as a polyhedral complex, contains
the boundary of the polytope $P$. In particular, the PL function defined by this tropical polynomial is the Legendre transform of the function $\psi$. You can see some
details of this in, say, Mikhalkin's paper <http://arxiv.org/pdf/math/0312530.pdf>,
in section 3. If you want to have the polytope $P$ itself appearing, then you
can take the tropical variety thought of as the graph of $f$, obtained by taking
the tropical polynomial $y+f$, where $y$ is an extra variable: see Proposition 3.5
in the above cited paper.


