The integer point transform as a complete invariant

The integer point transform $\sigma_\PP$ is an important invariant of a rational polytope $\PP$, and here we show that it complete invariant. We prove only necessary to evaluate at one algebraic in order uniquely determine by employing the Lindemann-Weierstrass theorem. Similarly, Fourier $\PP$ single point, $\PP$. identical uniqueness results also hold for cones. In addition, relating finite transforms, number \emph{integer evaluations} suffice give equivalent condition central symmetry set, terms transform, some facts about its local maxima. Most are proven arbitrary sets points $\R^d$.