  Let $M$ be a closed manifold and $\mathcal{A} \subseteq H^1_{\mathrm{dR}}(M)$
a polytope. For each $a \in \mathcal{A}$ we define a Novikov chain complex with
a multiple finiteness condition encoded by the polytope $\mathcal{A}$. The
resulting polytope Novikov homology generalizes the ordinary Novikov homology.
We prove that any two cohomology classes in a prescribed polytope give rise to
chain homotopy equivalent polytope Novikov complexes over a Novikov ring
associated to said polytope. As applications we present a novel approach to the
(twisted) Novikov Morse Homology Theorem and prove a new polytope Novikov
Principle. The latter generalizes the ordinary Novikov Principle and a recent
result of Pajitnov in the abelian case.
