  In four space-time dimensions, there are good theoretical reasons for
believing that General Relativity is the correct geometrical theory of gravity,
at least at the classical level. If one admits the possibility of extra
space-time dimensions, what would we expect classical gravity to be like? It is
often stated that the most natural generalisation is Lovelock's theory, which
shares many physical properties with GR. But there are also key differences and
problems. A potentially serious problem is the breakdown of determinism, which
can occur when the matrix of coefficients of second time derivatives of the
metric degenerates. This can be avoided by imposing inequalities on the
curvature. Here it is argued that such inequalities occur naturally if the
Lovelock action is obtained from Weyl's formulae for the volume and surface
area of a tube. Part of the purpose of this article is to give a treatment of
the Weyl tube formula in terminology familiar to relativists and to give an
appropriate (straightforward) generalisation to a tube embedded in Minkowski
space.
