  A multi-dimensional junction is obtained by identifying the boundaries of a
finite number of copies of an Euclidian half-space. The main contribution of
this article is the construction of a multidimensional vertex test function
G(x, y). First, such a function has to be sufficiently regular to be used as a
test function in the viscosity solution theory for quasi-convex Hamilton-Jacobi
equations posed on a multi-dimensional junction. Second, its gradients have to
satisfy appropriate compatibility conditions in order to replace the usual
quadratic pe-nalization function |x -- y| 2 in the proof of strong uniqueness
(comparison principle) by the celebrated doubling variable technique. This
result extends a construction the authors previously achieved in the network
setting. In the multi-dimensional setting, the construction is less explicit
and more delicate. Mathematical Subject Classification: 35F21, 49L25, 35B51.
