Quantitative Expressions for the Connectedness of Sets in ${\mathbb R}^n$

We prove that, for two arbitrary points $a$ and $b$ of a connected set $E\subset\nobreak {\mathbb R}^n$ ($n\ge 2$) and for any $\varepsilon>0$, there exist points $x_0=a$, $x_2,\dots,x_p=b$ in $E$ such that
$$\|x_1-x_0\|^n+\dots+\|x_p-x_{p-1}\|^n<\varepsilon.$$
We prove that the exponent $n$ in this assertion is sharp. The nonexistence of a chain of points in $E$ with
$$\|x_1-x_0\|^\alpha+\dots+\|x_p-x_{p-1}\|^\alpha<\varepsilon$$
for some $\alpha\in (1,n)$ proves to be equivalent to the existence of a nonconstant function $f\colon E\to {\mathbb R}$ in the class $\operatorname{Lip}_\alpha(E)$. For each such $\alpha$, we construct a curve $E(\alpha)$ of Hausdorff dimension $\alpha$ in ${\mathbb R}^n$ and a nonconstant function $f\colon E(\alpha)\to {\mathbb R}$ such that $f\in\operatorname{Lip}_\alpha(E(\alpha))$.