# Seeking a Weyl tube formula for Whitney stratified spaces

Background: Let $$X$$ be a smooth, compact Riemannian submanifold of euclidean space $$\mathbb{R}^n$$. H Weyl's tube formula asserts that for sufficiently small $$t > 0$$, the volume $$V(X;t)$$ of the radius-$$t$$ normal bundle around $$X$$ in $$\mathbb{R}^n$$ is given by a polynomial in $$t$$ of the form $$V(X;t) := \sum_{d=0}^n \lambda_i t^i,$$ where the $$\lambda_i$$ are intrinsic quantities, i.e., independent of the embedding $$X \hookrightarrow \mathbb{R}^n$$. Federer's work on curvature measures vastly extends this to the case where $$X \subset \mathbb{R}^n$$ is a compact set of positive reach. Unfortunately, very simple singular spaces, eg the shape of the letter $$X$$, do not have positive reach.

Question: Is there a Weyl-type tube formula for Whitney stratified spaces? Failing this, can we at least get good bounds in terms of the curvature measures of the strata?

Some leads: Theorem 3.4.2 of Adler and Taylor's book Topological Complexity of Smooth Random Functions appears promising at first glance; but it requires local convexity of $$X$$, which is again far too restrictive (eg, the singular point of the $$X$$-shape does not admit a convex neighbourhood). There's also a nice paper of Lotz here with a tube formula for real complete intersections. This is given in terms of degrees of defining polynomials, and hence does not directly apply to more general Whitney stratified spaces. Mather's work from Notes on topological stability describes systems of tubular neighbourhoods around strata as part of "control data", but I can't seem to get anything resembling volume bounds out of this. (See also this MO question, which is addressed by Federer's result).