Whitney regularity conditions are given in terms of limits of tangents and of secants, while Verdier regularity also involves limits of tangents. In the complex analytic world, Verdier regularity for a pair of strata (Y,Z) is equivalent to a condition on the fibres of a certain mapping (Henry-Merle, Teissier, Le-Teissier). From the complex projective point of view, a corollary of Zak's theorem on the tangent and secant varieties states for any irreducible algebraic projective variety X: the dimension of the singular set of X is at least the corank of the projective Gauss mapping of X minus one. In a very informal manner this means that if there is not much limits of tangent spaces at the singular set, then the singular set must be large.

I would like to address similar questions in the context of o-minimal geometries in an affine situation. Another way to look at this is in asking the following questions: 1 - Given a connected and enough differentiable and definable submanifold, what is the geometry of its boundary ? More simply how large is a the singular locus ? 2 - For a large class of functions I will prove such a result about the critical locus of a singular level.