  This is an extended example of the study of mirror symmetry via log schemes
and the discrete Legendre transform on affine manifolds, introduced by myself
and Bernd Siebert in "Mirror Symmetry via Logarithmic Degeneration Data I"
(math.AG/0309070). In this paper, I consider the construction as it applies to
the Batyrev-Borisov construction for complete intersections in toric varieties.
Given a pair of reflexive polytopes with nef decompositions dual to each other,
and given polarizations on the corresponding toric varieties, we construct
examples of toric degenerations and their dual intersection complexes, which
are affine manifolds with singularities. We show these affine manifolds are
related by the discrete Legendre transform, thus showing that the
Batyrev-Borisov construction is a special case of our more general
construction. The description of the dual intersection complexes in terms of
the combinatoricsof the setup generalises work of Haase and Zharkov in the
toric hypersurface case, and similar work of Ruan. In particular, this gives a
topological description of the Strominger-Yau-Zaslow fibrations on complete
intersections in toric varieties.
