  The convex hull property is the natural generalization of maximum principles
from scalar to vector valued functions. Maximum principles for finite element
approximations are often crucial for the preservation of qualitative properties
of the respective physical model. In this work we develop a convex hull
property for $\P_1$ conforming finite elements on simplicial non-obtuse meshes.
The proof does not resort to linear structures of partial differential
equations but directly addresses properties of the minimiser of a convex energy
functional. Therefore, the result holds for very general nonlinear partial
differential equations including e.g. the $p$-Laplacian and the mean curvature
problem. In the case of scalar equations the introduce techniques can be used
to prove standard discrete maximum principles for nonlinear problems. We
conclude by proving a strong discrete convex hull property on strictly acute
triangulations.
