  Pointwise tangential dimensions are introduced for metric spaces. Under
regularity conditions, the upper, resp. lower, tangential dimensions of X at x
can be defined as the supremum, resp. infimum, of box dimensions of the tangent
sets, a la Gromov, of X at x. Our main purpose is that of introducing a tool
which is very sensitive to the "multifractal behaviour at a point" of a set,
namely which is able to detect the "oscillations" of the dimension at a given
point. In particular we exhibit examples where upper and lower tangential
dimensions differ, even when the local upper and lower box dimensions coincide.
Tangential dimensions can be considered as the classical analogue of the
tangential dimensions for spectral triples introduced in math.OA/0202108 and
math.OA/0404295, in the framework of Alain Connes' noncommutative geometry.
