TITLE: If Hausdorff distance $d(A,B) < ε$ for some $ε$, where $A,B$ of $\mathbb{R}^n$ are compact, convex subsets does $\dim(A) = \dim(B)$?
QUESTION [0 upvotes]: full question:
The Hausdorff distance between two convex, compact subsets $A$ and $B$ of $\mathbb{R}^n$ is defined as:
$$d(A,B) = \max \{ \max \operatorname{dist}(a,B), \max \operatorname{dist}(b,A) \mid b\in B \text{ and } a \in A\}$$
Prove or disprove that there exists $\varepsilon >0$ such that if $d(A,B) < \varepsilon$ then $\dim A = \dim B$ ?
I'm not really sure what does the dimensions of $A$ and $B$ have to do with distance between the two convex sets in this case, we know they are both subsets of $\mathbb{R}^n$, I'd really appreciate any help.

REPLY [0 votes]: A space-filling curve ( see https://en.wikipedia.org/wiki/Space-filling_curve ) is constructed as the limit of a sequence of straight-segmented finite chains. The sequence of the (1-dimensional) chains converges (under Hausdorff distance) to a square, a 2-dimensional set. This disproves the statement in the question.
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