Introduction
Let $K$ be a nice closed domain in $\mathbb{R}^2$, for example the closed unit ball. Recall that the Hausdorff distance on the family $C(K)$ of nonempty compact subsets of $K$ is defined as
$$ D(A,B) = \inf\{ \delta>0: A\subset B_\delta \text{ and } B\subset A_\delta \}, $$
where $X_\delta=\{y:\text{dist}(y,X)<\delta \}$ is the $\delta$ neighborhood of $X$.
Now consider the restriction of $D$ to the family $A_n$ of nonempty intersections of real algebraic curves of degree at most $n$ with $K$. Intuitively, it seems clear to me that the metric space $(A_n,D)$ should be finite dimensional, in any reasonable sense; for example, it should have finite upper box dimension, and even be doubling (meaning that a ball of radius $2\delta$ is covered by a uniformly bounded number of balls of radius $\delta$). However, so far I've failed to find a proof.
Question:
Is the space $(A_n,D)$ doubling/of finite box counting dimension? (the latter means that $A_n$ can be covered by $\delta^{-C}$ $D$-balls of radius $\delta$ for some $C>0$). If so, what is its box counting dimension? (in other words, what is the best $C$?).
Some remarks:
An obvious approach is to estimate the Hausdorff distance in terms of the coefficients, but this runs into several problems; for example, there are points of discontinuity: take $P_{\varepsilon}(x)$ with a single real root at $x=0.5$ and such that $P(\varepsilon)=\varepsilon$, and let $Q_\epsilon(x,y)=P_\epsilon(x)$. In the limit $\epsilon=0$, a large new component for $Q_\epsilon=0$ is created.
Another natural approach is to fix a large number $N$ (at least $n^2+1$) and consider the map that sends $N$-tuples of points which are at pairwise distances at least $\delta>0$ and which lie on some algebraic curve, to the algebraic curve passing through all of them. If this function was Lipschitz, or even Hölder (from the standard metric on $\mathbb{R}^{2N}$ to $(A_n,d)$), it would follow that the latter is doubling.
This is related to effective versions of Łojasiewicz inequality. It is known that for each $n$ there exists $\alpha$, such that for any nonempty real algebraic curve $\gamma$ of degree $n$, if $|P(x)|<\varepsilon$, then $\text{dist}(x,\gamma)\le C \varepsilon^\alpha$, where the constant $C$ depends on the curve. See for example this nice article by Johnson and Kollár.
If $C$ was a continuous functions of the coefficients, this would easily imply that $(A_n,D)$ is doubling, but $C$ is not continuous in general, as can be seen from the same example as in the first remark.
We can prove (with an elementary argument) the following weaker bound: $A_n$ can be covered by $\delta^{O(\log\delta)}$ balls of radius $\delta$ (in the Hausdorff metric). This is enough for our application.
Of course, one may ask this question for more general real algebraic varieties, and also for complex varieties (the comple case should be easier).
Motivation
If $\mathcal{F}$ is a class of curves in $\mathbb{R}^2$, a tube of width $\delta$ is the $\delta$-neighborhood of a curve in $\mathcal{F}$. A set $E\subset\mathbb{R}^2$ is then called tube-null (with respect to $\mathcal{F}$) if for every $\delta>0$ it can be covered by countably many tubes so that the sum of their widths is at most $\delta$.
Together with Ville Suomala we prove that if $\mathcal{F}$ is the class of all lines, then there are sets of Hausdorff dimension 1 which are not tube-null (this answers a question of Carbery, Soria and Vargas). The question arose in trying to generalize this result to the family of algebraic curves of degree at most $n$.
