  We prove bounds for the volume of neighborhoods of algebraic sets, in the
euclidean space or the sphere, in terms of the degree of the defining
polynomials, the number of variables and the dimension of the algebraic set,
without any smoothness assumption. This generalizes previous work of Lotz on
smooth complete intersections in the euclidean space and of B\"urgisser, Cucker
and Lotz on hypersurfaces in the sphere, and gives a complete solution to
Problem 17 in the book titled "Condition" by B\"urgisser and Cucker.

Algebraic Geometry;Numerical Analysis;Numerical Analysis;Optimization and Control;Probability