  We introduce a method that uses the Cauchy-Crofton formula and a new
curvature formula from integral geometry to reweight the sampling probabilities
of Metropolis-within-Gibbs algorithms in order to increase their convergence
speed. We consider algorithms that sample from a probability density
conditioned on a manifold $\mathcal{M}$. Our method exploits the symmetries of
the algorithms' isotropic random search-direction subspaces to analytically
average out the variance in the intersection volume caused by the orientation
of the search-subspace with respect to the manifold $\mathcal{M}$ it
intersects. This variance can grow exponentially with the dimension of the
search-subspace, greatly slowing down the algorithm. Eliminating this variance
allows us to use search-subspaces of dimensions many times greater than would
otherwise be possible, allowing us to sample very rare events that a
lower-dimensional search-subspace would be unlikely to intersect.
  To extend this method to events that are rare for reasons other than their
support $\mathcal{M}$ having a lower dimension, we formulate and prove a new
theorem in integral geometry that makes use of the curvature form of the
Chern-Gauss-Bonnet theorem to reweight sampling probabilities. On the side, we
also apply our theorem to obtain new theoretical bounds for the volumes of real
algebraic manifolds.
  Finally, we demonstrate the computational effectiveness and speedup of our
method by numerically applying it to the conditional stochastic Airy operator
sampling problem in random matrix theory.
