Quantitative expressions for the connectedness of sets in ℝ n

Abstract

We prove that, for two arbitrary points a and b of a connected set E ⊂ Rn (n ≥ 2) and for any e > 0, there exist points x0 = a, x2,..., xp = b in E such that $$\parallel {x_1} - {x_0}{\parallel ^n} + \cdots + \parallel {x_p} - {x_{p - 1}}{\parallel ^n} < \varepsilon$$ . We prove that the exponent n in this assertion is sharp. The nonexistence of a chain of points in E with $$\parallel {x_1} - {x_0}{\parallel ^\alpha } + \cdots + \parallel {x_p} - {x_{p - 1}}{\parallel ^\alpha } < \varepsilon$$ for some α ∈ (1, n) proves to be equivalent to the existence of a nonconstant function f: E → R in the class Lipα(E). For each such α, we construct a curve E(α) of Hausdorff dimension α in Rn and a nonconstant function f: E(α) → R such that f ∈ Lipα(E(α)).