This problem was motivated by the MO problems: "Running most of the time in a connected set", "Is every metric continuum almost path connected?" and "Are $\varepsilon$-connected components dense?".

Let $p$ be a positive real number. A metric space $(X,d)$ is called *$\ell_p$-chain connected* if for every points $x,y\in X$ and $\varepsilon>0$ there exists a sequence $C_0,\dots,C_n$ of connected subsets of $X$ of diameter $<\varepsilon$ such that $x\in C_0$, $y\in C_n$ and $\sum_{i=1}^n d(C_{i-1},C_i)^p<\varepsilon$. Here $d(A,B)=\inf\{d(a,b):a\in A,\;b\in B\}$.

**Problem 1.** Is each connected compact subset of a Euclidean space $\mathbb R^n$ $\ell_1$-chain connected? What is the answer for $n=2$?

**Remark 1.** By the method of the proof of Lemma from the answer to this MO-problem, it can be shown that each connected compact subspace of $\mathbb R^n$ is $\ell_p$-chain connected for any $p>n-1$.

**Problem 3.** Is each connected compact metric space $\ell_1$-chain connected? $\ell_p$-connected for some $p$?

**Remark 2.** Each $\ell_1$-chain connected compact metric space is almost path connected in the sense of this MO problem.

**Added in Edit.** It seems that the the answer of Anton Petrunin to this problem also yields a counterexample to Problem 3.