Let $P\subset\mathbb{R}^{n}$ be a lattice polytope (vertices are in $\mathbb{Z}^{n}$). Set $P_{k}=P\cap k^{-1}\mathbb{Z}^{n}$ for $k\geq1$. Given a concave function $\phi:P\rightarrow\mathbb{R}$ (continuous on $P$). My question is: whether we have two-terms Euler-Maclaurin asymptotics \begin{equation} \sum_{u\in P_{k}}\phi(u)=\int_{P}\phi\ \mathrm{d}x\cdot k^{n}+\frac{1}{2}\int_{\partial P}\phi\ \mathrm{d}\sigma\cdot k^{n-1}+\mathrm{o}(k^{n-1})\ ?\label{eq:expansion} \end{equation} Where $\mathrm{d}x$ is Lebesgue measure and the boundary measure $\mathrm{d}\sigma$ is defined as follows.
For each facet $F\subset\partial P$, let $A_{F}\subset\mathbb{R}^{d}$ be the affine subspace generated by $F$, then $\mathrm{d}\sigma|_{F}$ is the Lebesgue measure normalized by the lattice $A_{F}\cap\mathbb{Z}^{d}$. For example, $P$ is the convex hull of $(0,0),(1,0),(0,2)$, then the length (measured by $\mathrm{d}\sigma$) of the hypotenuse is $1$ (not $\sqrt{5}$).
