# Representing integer points inside a polytope using a unit hypercube

Let $D\subset\mathbb{Z}\times\mathbb{Z}$ such that $D$ is formed by the points bounded by a convex polytope. Let $f:D\to\mathbb{R}$ defined by $f(x,y)=ax+by+c$, where $a,b,c\in\mathbb{R}$ and $g:D\to\mathbb{N}$.

Conjecture: We can always find a subset $\tilde{D}\subset D$, such that $\tilde{D}$ is a subset of some unit square (or maybe some other small set?) and a function $\tilde{g}:\tilde{D}\to\mathbb{N}$ such that $$\sum_{(x_i,y_i)\in D}g(x_i,y_i)f(x_i,y_i)=\sum_{(x_i,y_i)\in \tilde{D}}\tilde{g}(x_i,y_i)f(x_i,y_i)$$

Is this true considering $D\subset\mathbb{Z}^n$ and $\tilde{D}$ a subset of some unit hypercube?

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I think D containing the points $(-6,1), (2,0), (5, -2)$ may be an illustrative example, as adding the two extreme points gives $c+c-a-b$, while $D'$ can contain at most one of the points of D. Gerhard "Ask Me About System Design" Paseman, 2011.08.12 –  Gerhard Paseman Aug 12 '11 at 20:28