  We derive exponential bounds on probabilities of large deviations for "light
tail" martingales taking values in finite-dimensional normed spaces. Our
primary emphasis is on the case where the bounds are dimension-independent or
nearly so. We demonstrate that this is the case when the norm on the space can
be approximated, within an absolute constant factor, by a norm which is
differentiable on the unit sphere with a Lipschitz continuous gradient. We also
present various examples of spaces possessing the latter property.
