Доказывается, что для любых двух точек $a$ и $b$ связного множества $E\subset{\mathbb R}^n$ ($n\geqslant 2$) и для любого $\varepsilon>0$ в $E$ найдутся такие точки $x_0=a$, $x_2,…,x_p=b$, что $$ \|x_1-x_0\|^n+…+\|x_p-x_{p-1}\|^n<\varepsilon. $$ Доказывается, что показатель $n$ в этом утверждении уменьшить нельзя. Невозможность выбрать во множестве $E$ указанную цепочку точек с $$ \|x_1-x_0\|^\alpha+…+\|x_p-x_{p-1}\|^\alpha<\varepsilon $$ для некоторого $\alpha\in (1,n)$ оказывается эквивалентной существованию непостоянной функции $f\colon E\to {\mathbb R}$ из класса $\operatorname{Lip}_\alpha(E)$. Для каждого такого $\alpha$ в ${\mathbb R}^n$ строится такая кривая $E(\alpha)$ хаусдорфовой размерности $\alpha$ и такая непостоянная функция $f\colon E(\alpha)\to {\mathbb R}$, что $f\in\operatorname{Lip}_\alpha(E(\alpha))$. Библиография: 3 названия.

We prove that, for two arbitrary points a and b of a connected set E ⊂ R n ( n ≥ 2) and for any ε > 0, there exist points x 0 = a, x 2 ,..., x p = b in E such that ∥x_1 - x_0∥ ^n + ⋯ + ∥x_p - x_p - 1∥ ^n < ε . 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∥ ^α + ⋯ + ∥x_p - x_p - 1∥ ^α < ε for some α ∈ (1, n ) proves to be equivalent to the existence of a nonconstant function f : E → R in the class Lip α ( E ). For each such α, we construct a curve E (α) of Hausdorff dimension α in R n and a nonconstant function f : E (α) → R such that f ∈ Lip α ( E (α)).

Том 92 выпуск 6 декабрь 2012 УДК 517.54 Об оценке длин лемнискат О. Н. Косухин В работе при любом натуральном n и любом C > 0 получена интегральная формула для вычисления длин |L(Pn, C)| лемнискат L(Pn, C) := {z : |Pn(z)| = C} алгебраических многочленов Pn(z) := z n + cn-1z n-1 + • •

For any natural number n and any C > 0, we obtain an integral formula for calculating the lengths | L ( P n , C )| of the lemniscates L( P_n ,C): = {z:| P_n ( z )| = C} of algebraic polynomials P n ( z ):= z n + c n −1 z n −1 + ... + c 0 in the complex variable z with complex coefficients c j , j = 0, ..., n − 1, and establish the upper bound for the quantities λ _n : = sup{| L( P_n ,1)|:P_n (z)}, which is currently best for 3 ≤ n ≤ 10 14 . We also study the properties of the derivative S ′( C ) of the area function S ( C ) of the set z : | P n ( z )| ≤ C .

The following new geometric criterion is proved: a real Banach space ( X , ‖ · ‖) is a Hilbert space if and only if for any three points A,B,C of this space not belonging to a line there are three altitudes in the triangle ABC intersecting at one point.

As proved by Hilbert, it is, in principle, possible to construct an arbitrarily close approximation in the Hausdorff metric to an arbitrary closed Jordan curve Γ in the complex plane {z} by lemniscates generated by polynomials P(z). In the present paper, we obtain quantitative upper bounds for the least deviations H n (Γ) (in this metric) from the curve Γ of the lemniscates generated by polynomials of a given degree n in terms of the moduli of continuity of the conformal mapping of the exterior of Γ onto the exterior of the unit circle, of the mapping inverse to it, and of the Green function with a pole at infinity for the exterior of Γ. For the case in which the curve Γ is analytic, we prove that H n (Γ) = O(q n ), 0 ≤ q = q(Γ) < 1, n → ∞.