Let $P$ be polytope in $\mathbb{R}^d$ with vertices in $\mathbb{Z}^d$ and let $h(z_1,\ldots,z_d)\in\mathbb{Z}[z_1,\ldots,z_d]$ be a degree one polynomial with integer coefficients, thought of as a "height function'' on $\mathbb{Z}^d$. For each integer $k\in\mathbb{Z}$ consider the lattice points in the cross-section of height $k$: $$P_k:=P\cap \{\mathbf{x}\in\mathbb{Z}^d: h(\mathbf{x})=k\}.$$ Suppose that $|P_k|=0$ for $k<m$ and $k>n$ for some integers $m<n$. I am interested in the integer sequence: $$|P_m|, |P_{m+1}|, \ldots,|P_n|.$$
Question: Have these sequences been studied? I am particularly interested in cases when the sequence is unimodal. This would be a discrete analogue of the unimodality of cross-sectional volume, proved in this paper using the Brunn-Minkowski theorem: