For relatively open convex polyhedra (cells)Q ⊂ ℝd we put χ(Q):=(−1)dimQ. Any polyhedronQ ⊂ ℝd is the disjoint union of a finite number of cells:\(P = \bigcup\limits_i {Q_i } \). We show that\(\chi (P): = \sum\limits_i \chi (Q_i )\) is independent of the specific decomposition ofP into disjoint cells and therefore is uniquely determined byP. Since every closed convex polyhedron is the disjoint union of its relatively open faces of all dimensions, χ(P) is the Euler characteristic ofP. We finally present a new and elementary proof of the theorem of Euler-Schlafli.