  We consider multi-polytopes to describe non-Fano toric varieties and their
associated anticanonical Calabi-Yau hypersurfaces. From the periods of the
mirror manifold the $\widehat{\Gamma}$-conjecture is shown to hold for examples
of Calabi-Yau hypersurfaces in non-Fano ambient spaces, extending earlier work
by Abouzaid et al by employing a generalized Duistermaat-Heckman measure.
