$(r)$ does not imply $(n)$ or $(npf)$ for definable sets in non polynomially bounded o-minimal structures

Abstract

It is known that if two subanalytic strata satisfy Kuo's ratio test, then the normal cone of the larger stratum $Y$ along the smaller $X$ satisfies two nice properties: the fiber of the normal cone at any point of $X$ is the tangent cone to the fiber of $Y$ over that point; the projection of the normal cone to $X$ is open ("normal pseudo-flatness"). We present an example with $X$ a line and $Y$ a surface which is definable in any non polynomially bounded o-minimal structure such that the pair satisfies Kuo's ratio test, but neither of the above properties hold for the normal cone.