  We prove a general theorem providing smoothed analysis estimates for conic
condition numbers of problems of numerical analysis. Our probability estimates
depend only on geometric invariants of the corresponding sets of ill-posed
inputs. Several applications to linear and polynomial equation solving show
that the estimates obtained in this way are easy to derive and quite accurate.
The main theorem is based on a volume estimate of \epsilon-tubular
neighborhoods around a real algebraic subvariety of a sphere, intersected with
a disk of radius \sigma. Besides \epsilon and \sigma, this bound depends only
the dimension of the sphere and on the degree of the defining equations.
