Given a codimension 1 smooth immersed submanifold \(M^ m\) in \(E^{m+1}\), the authors study the structure of the frontier H(f) of the convex hull of the image of \(M^ m\) in \(E^{m+1}\). First of all, they show that there exists a star-shaped smooth embedding h of the m-sphere \(S^ m\) into \(E^{m+1}\) with \(H(h)=H(f)\). Using this, a panel structure of H(f) is introduced which is related to the decomposition of a convex body into ''contact sets''. The panel structure is also shown to be well behaved for a residual set of the space of smooth embeddings. As a generalization of the Euler relation for polyhedra, they obtain a relation between the Euler numbers of panels.