Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Let $X$ be a ${\mathcal H}^m$-rectifiable subset of ${\bf R}^n$ and $x = \lim_i x_i$, where $x_i$ and $x$ are points possessing approximate tangent spaces.

Question Does it follow that $T_{x_i}X$ converges to $T_xX$? Or (in case it is not true in general): Under what assumptions does it hold?

share|improve this question

1 Answer 1

In general this can fail for all points: let $\{q_i\}$ be a countable dense set in the plane, and let $$X= \bigcup_i S(q_i,2^{-i}),$$ where $S(q,r)$ is the circle of center $q$ and radius $r$. Then $X$ is $1$-rectifiable, yet for all $x\in X$ and all $\delta>0$ you can find points in the ball $B(x,\delta)$ with tangent lines very different from $T_x X$ (in fact, you can find a dense set of tangent lines inside any such ball).

You can create a similar example with $X$ made up of a dense set of segments pointing in a dense set of directions inside any ball, and you can easily replicate this construction for any $n,m$.

Since stability under countable unions is a key feature of the notion of rectifiability, I doubt you can find a useful setting under which your statement holds.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.