This paper is concerned with the definition of shape functions for high-order Whitney finite element spaces. The main goal in the present paper is to study higher-order Whitney complexes and to design shape functions for Whitney
-elements that use only dofs associated with integration subdomains of dimension
. This task is performed with the introduction of so-called small simplices, which are defined by means of a particular homothety. Degrees of Freedom are then the integrals over suitable
-dimensional subdomains, linear combinations of small
-simplices. It is also provided a basis for these elements on simplicial meshes and is given a geometrical localization of all degrees of freedom.