In the space \(E^3\) the closed curve \(X\) lying on its convex hull is considered, \(X\) need not lie on the boundary of the convex hull. A point where the torsion \(\tau\) or the curvature \(k\) changes sign is called a vertex or an inflection, respectively.
A point where \(\frac\tau k\) or \(\frac k\tau\) has a local strong extremum is called a Darboux vertex. The numbers of vertices, inflections and Darboux vertices is denoted by \(V\), \(K\), \(D\), respectively.
Together with the curve \(X\) its tangent image \(T=X'\) as a curve on the unit sphere \(S^2\) is considered. The vertex theorem \(\Sigma:=V+K+D\geq 4\) for a closed space curve \(X\) is proved. Some special properties are verified in the case \(\Sigma=4\). Especially the vertex theorem for some curves on the torus and for spherical curves are studied. The geometric interpretation of Darboux vertices is found.