A problem I'm on a hunt to figure out. I'm not sure whether it's better placed here or on Overflow- it has the feeling of being very standard for the right person but I haven't done enough geometric measure theory to spot it. Happy to be prompted to migrate if it seems beyond scope.
Setup: Let $f \in \mathbb{R}[x_1,\dots,x_N]$ and assume the real part of the corresponding algebraic set $S$ defined as $\mathcal{V}(f)\cap\mathbb{R}^N$ is compact. If it helps, assume that it's smooth. Let $\epsilon > 0$ and suppose $x\in S$, i.e. $f(x) = 0$.
Question: Let $B(x,\epsilon)$ be the Euclidean ball of radius $\epsilon$ around $x$ in $\mathbb{R}^N$. Can we associate some geometric condition number $c(S)$ based on a well-known geometric quantity (e.g. extrinsic curvature) to $S$ and give a lower bound on the (N-1)-dimensional Hausdorff measure (in $\mathbb{R}^N$) of the set $ S \cap B(x,\epsilon)$ in terms of $c(S)$?
This is related to a probability question: Given $x$ and $\epsilon$, is there a somewhat satisfying lower bound on the probability you get a point within $\epsilon$ of $x$ when you sample from the uniform distribution on $S$ given by Hausdorff measure? It seems visually intuitive that this should depend on extrinsic curvature, but I can't quite get it there.
An upper bound would also be nice, but that's going to be trickier and rely on the algebraic properties more (you need Bezout's Theorem at least to limit spirals, I think).
The background: Diaconis et al. have a type of tutorial on these questions https://statweb.stanford.edu/~cgates/PERSI/papers/sampling11.pdf. In particular, Section 2.2 recalls the following area formula recorded in Federer's Geometric Measure Theory. It gives a formula for the Hausdorff measure of a set in terms of a parameterization. The following is a slightly weaker form of the result for conciseness.
Theorem Let $\lambda^m$ denote the Lebesgue measure on $\mathbb{R}^m$. If $u:\mathbb{R}^m\to\mathbb{R}^n$ is Lipschitz, $A\subseteq\mathbb{R}^m$ is $\lambda^m$ measurable, $u(A) = P$, and $u$ is injective: $$ H^m(P) = \int_{\mathbb{R}^n} 1_P(y) H^m(dy) = \int_A J_m u(z) \lambda^m(dz) $$ where $J_m u(z)$ is the Jacobian $\sqrt{\det((Du(z)^T Du(z))}$.
In our context $P$ is our set $S\cap B(x,\epsilon)$ and we a priori don't know a parameterization $u:\mathbb{R}^{N-1}\to\mathbb{R}^N$. The Implicit Function Theorem is standard for this, with the caveat that $\epsilon$ must be small enough. Assume for the sake of convenience that we can use the IFT on the last variable. We at least know that if $\epsilon$ is small enough we have a parameterization $u:U\to\mathbb{R}^N$ with $U\subseteq\mathbb{R}^{N-1}$ of the form $u(z) = (z,g(z))$ for some $g:\mathbb{R}^{N-1}\to\mathbb{R}$. Furthermore, let $B=\frac{\partial f}{\partial x_N}(u(z)))^{-1}$. Then $Du(z)$ is a matrix of the form $(I_{n-1} | (-B\hat\nabla_f(u(z)))^T$ where $\hat\nabla_f(u(z))$ is the first $n-1$ coordinates of $\nabla_f(u(z))$. So $Du(z)^T Du(z) = I_{n-1} + B^2 \hat\nabla_f(u(z))\hat\nabla_f(u(z))^T$.
That's the best I have for now: There's probably some better way to write that matrix for the determinant, or even a better approach to the question. It still seems like there's more say after that if it's related to e.g. extrinsic curvature or some other familiar quantity.