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This question is inspired by this other question of mine. Is there an example of rectifiable Jordan curve $\gamma:[0,1]\to\mathbb R^2$ such that, for some $\delta>0$, no polygonal path $\overline{\gamma(t_0)\gamma(t_1)\cdots\gamma(t_n)}$ with $0=t_0<t_1<\cdots<t_n=1$ and $\max_{1\leq i\leq n}(t_i-t_{i-1})<\delta$, is a Jordan curve? According to this question (with no answer yet), such curve $\gamma$ cannot be piecewise regular.

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Matemáticos Chibchas
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1 Answers1

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Lemma 2 in the paper Tverberg answers your question and the paper answers your first post, I think.

Gio67
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