Structure of solutions to continuous constraint satisfaction problems through the statistics of wedged and inscribed spheres

The study of random landscapes has long relied on counting stationary points: metastable states and the barriers between them. However, this method is useless for describing flat regions, common in constraint satisfaction problems. We introduce a characterization of flat regions by counting the number of spheres that can be uniquely inserted into them, either by wedging spheres of fixed radius or by inscribing spheres of variable radius. The ratio of these counts constrains the topology of the solution space. We apply this characterization to the spherical perceptron and show the existence of at least two topological regimes.