A Property of a Continuous Curve in Euclidean Space
Abstract Generated abstract
This note studies an operation on continuous curves in Euclidean space that replaces selected open subintervals of a parametrization by the corresponding straight line segments between their endpoint values. Using the characteristic of a continuous function, expressed through maximal delta partitions, the paper proves several invariance and monotonicity lemmas under reparametrization and linear replacement. These results are applied to show that if a finite sequence of such replacements produces successively distinct curves, then the final curve cannot coincide with the initial one. The argument is first given for one-dimensional Euclidean space and is then indicated for higher dimensions by projection.
Full Text
A Property of a Continuous Curve in Euclidean Space
L. E. Portnov
(Presented by Prof. P. P. Kufarev)
Let \(L_0\) be a continuous curve in the Euclidean space \(E_1\), and let \(\{f(t)\}\) be a family of equivalent mappings [1] defining \(L_0\).
Let \(f_0(t)\in\{f(t)\}\).
Take on \([a,b]\) an open set
\[ G=\bigcup_i (a_i,b_i), \]
where \((a_i,b_i)\), \(i=1,2,\ldots\), are the component intervals, and put
\[ \overline{f_0}(t)= \begin{cases} f_0(t), & \text{for } t\in [a,b]\setminus\displaystyle\bigcup_i(a_i,b_i),\\[6pt] \dfrac{f(a_i)(b_i-t)+f(b_i)(t-a_i)}{b_i-a_i}, & \text{for } t\in [a_i,b_i]. \end{cases} \]
Let now \(L_1\) be the curve determined by the mapping \(\overline{f_0}(t)\), and let \(\{f_1(t)\}\) be the family of equivalent mappings containing the function \(\overline{f_0}(t)\).
Let \(f_1(t)\in\{f(t)\}\).
Take on \([a,b]\) an open set
\[ G=\bigcup_i (c_i,d_i), \]
where \((c_i,d_i)\), \(i=1,2,\ldots\), are the component intervals, and put
\[ \overline{f_1}(t)= \begin{cases} f_1(t), & \text{for } t\in [a,b]\setminus\displaystyle\bigcup_i(c_i,d_i),\\[6pt] \dfrac{f(c_i)(d_i-t)+f(d_i)(t-c_i)}{d_i-c_i}, & \text{for } t\in [c_i,d_i]. \end{cases} \]
Let \(\{f_2(t)\}\) be the family of equivalent mappings containing the function \(\overline{f_1}(t)\), and let \(L_2\) be the curve determined by this family.
Continuing this process, we obtain the curves
\[ L_0,L_1,\ldots,L_n, \tag{1} \]
and the corresponding families of equivalent mappings
\[ \{f(t)\},\quad \{f_1(t)\},\ldots,\{f_n(t)\}, \tag{2} \]
some parametric representations of curves
\[ \overline{f_0}(t),\ \overline{\overline{f_0}}(t),\ \overline{f_1}(t),\ldots,\overline{f_{n-1}}(t) \tag{3} \]
and open sets
\[ G,\ G_1,\ G_2,\ldots,G_n. \tag{4} \]
Theorem. If the sequences (3) and (4) are such that
\[ L_0 \ne L_1 \ne L_2 \ne \ldots \ne L_n, \]
then
\[ L_0 \ne L_n. \]
To prove the theorem we shall establish some properties of the characteristic of a continuous function [2].
Lemma 1. If the function \(\varphi(t)\), \(t \in [a,b]\), is continuous and nondecreasing, then for any continuous function \(f(t)\), \(t \in [\varphi(a),\varphi(b)]\), and any \(\delta > 0\) we have
\[ N[f(t),\delta;\varphi(a),\varphi(b)] = N\{f[\varphi(t)],\delta;a,b\}. \]
Proof. Let
\[ \varphi(a)=t_0<t_1<\ldots<t_{N_1}=\varphi(b). \tag{6} \]
be a \(\delta\)-partition of \([\varphi(a),\varphi(b)]\) for \(f(t)\) such that
\[ N_1=N[f(t),\delta;\varphi(a),\varphi(b)], \]
and, analogously,
\[ a=s_0<s_1<\ldots<s_{N_2}=b \tag{7} \]
be a \(\delta\)-partition of \([a,b]\) for \(f[\varphi(t)]\) such that
\[ N_2=N\{f[\varphi(t)],\delta;a,b\}. \]
It is clear that
\[ \varphi(s_0)<\varphi(s_1)<\ldots<\varphi(s_{N_2}) \tag{8} \]
will be a \(\delta\)-partition for \(f(t)\). Hence,
\[ N_2 \leq N_1. \tag{9} \]
Let \(t_i' \in \varphi^{-1}(t_i)\); then
\[ t_0'<t_1'<\ldots<t_{N_1}' \tag{10} \]
will be a \(\delta\)-partition for \(f[\varphi(t)]\), and, consequently,
\[ N_2 \geq N_1. \tag{11} \]
From (9) and (11) we have
\[ N_1=N_2. \]
Corollary. If the functions \(f_1(t)\) and \(f_2(t)\), \(t \in [a,b]\), are equivalent, then for any \(\delta>0\)
\[ N[f_1(t),\delta;a,b]=N[f_2(t),\delta;a,b]. \]
Proof. Since the functions \(f_1(t)\) and \(f_2(t)\) are equivalent, there exist continuous nondecreasing functions \(\varphi(t)\), \(\psi(t)\), \(t \in [a,b]\), such that
\[ \varphi(a)=\psi(a)=a,\qquad \varphi(b)=\psi(b)=b. \]
and
\[ f_1[\varphi(t)] = f_2[\psi(t)]. \tag{12} \]
By Lemma 1, taking (12) into account, we have
\[ N[f_1(t),\delta;a,b]=N\{f_2[\psi(t)],\delta;a,b\}=N[f_2(t),\delta;a,b], \]
that is,
\[ N[f_1(t),\delta;a,b]=N[f_2(t),\delta;a,b]. \]
Lemma 2. For a continuous function \(f(t)\), \(t\in [a,b]\), and a partition of the interval \([a,b]\)
\[ a=t_0<t_1<\ldots<t_\kappa=b \]
for any \(\delta>0\),
\[ \sum_{i=1}^{\kappa} N_i \leq N[f(t),\delta; a,b]\leq \sum_{i=1}^{\kappa}N_i+\kappa-1, \tag{13} \]
where
\[ N_i=N[f(t),\delta; t_{i-1},t_i]\quad i=1,2,\ldots,\kappa . \]
Proof. Let
\[ S^i_0<S^i_1<\ldots<S^i_{N_i}\qquad i=1,2,\ldots,\kappa \tag{14} \]
be \(\delta\)-partitions determining the \(\delta\)-characteristics. The totality (14) determines on \([a,b]\) a \(\delta\)-partition for the function \(f(t)\), and therefore
\[ \sum_{i=1}^{\kappa}N_i \leq N[f(t);\delta; a,b]. \tag{15} \]
Suppose that the right-hand part of inequality (13) is not satisfied, that is,
\[ N[f(t),\delta; a,b]\geq \sum_{i=1}^{\kappa}N_i+\kappa . \]
Then the \(\delta\)-partition giving \(N[f(t),\delta; a,b]\) determines, on at least one interval \([t_{i-1},t_i]\), a \(\delta\)-partition consisting of more than \(N_i\) intervals, which contradicts the definition of \(N_i\). Consequently,
\[ N[f(t),\delta; a,b]\leq \sum_{i=1}^{\kappa}N_i+\kappa-1. \tag{16} \]
From (15) and (16), (13) follows.
Lemma 3. If \(f(t)\), \(t\in [a,b]\), is a continuous function and
\[ g(t)= \begin{cases} f(t), & \text{for } t\in [a,b]\setminus \displaystyle\bigcup_i(a_i,b_i),\\[6pt] \dfrac{f(a_i)(b_i-t)+f(b_i)(t-a_i)}{b_i-a_i}, & \text{for } t\in (a_i,b_i), \end{cases} \]
where \(\displaystyle \bigcup_i(a_i,b_i)=G\subset [a,b]\), and \((a_i,b_i)\), \(i=1,2,\ldots\), are the component intervals of the set \(G\), then
\[ N[f(t),\delta; a,b]\geq N[g(t),\delta; a,b]. \]
Proof. Let
\[ a=t_0<t_1<\ldots<t_m=b, \tag{17} \]
be a \(\delta\)-partition determining
\[ N[g(t),\delta; a,b]=m. \]
Take the partition
\[ a=p_0<p_1<\ldots<p_m=b, \tag{18} \]
where, in the case \(f(t_i)=g(t_i)\), \(p_i=t_i\).
Since
\[ \omega_{[t_{i-1},\,t_i]} f(t)\geq \omega_{[t_{i-1},\,t_i]} g(t), \]
then
\[ N[f(t),\delta; a,b]\geq N[g(t),\delta; a,b]. \]
Let now \(t_z\) be the first point from (17) for which
\[ f(t_z)\ne g(t_z). \]
It is clear that \(t_z\in (a_i,b_i)\subset G\).
Since \(f(a_i)=g(a_i)\), \(f(b_i)=g(b_i)\), and \(g(t)\) is monotone on \([a,b]\), there exists \(p_z\in [a_i,b_i]\) such that
\[ f(p_z)=g(t_z). \]
Let
\[ \varphi(t)= \begin{cases} t, & \text{for } t\in [t_{z-1},a_i],\\[6pt] \dfrac{a_i(t_z-t)+p_z(t-a_i)}{t_z-a_i}, & \text{for } t\in [a_i,t_z]. \end{cases} \]
Since \(\varphi(t)\) is monotone, we have
\[ \omega_{[t_{z-1},p_z]} f(x)=\omega_{[z-1,t_z]} f[\varphi(t)], \]
but
\[ \omega_{[t_{z-1},t_z]} f[\varphi(t)]\geq \omega_{[t_{z-1},t_z]} g(t). \]
Consequently,
\[ \omega_{[p_{z-1},p_z]} f(t)\geq \omega_{[t_{z-1},t_z]} g(t). \]
Now put \(p_{z+1}=t_{z+1}\) when \(f(t_{z+1})=g(t_{z+1})\), while if \(f(t_{z+1})\ne g(t_{z+1})\), choose \(p_{z+1}\) in the same way as \(p_z\). Thus, to each \([t_{i-1},t_i]\), \(i=1,2,\ldots,m\), we assign \([p_{i-1},p_i]\), and moreover
\[ \omega_{[p_{i-1},p_i]} f(t)\geq \omega_{[t_{i-1},t_i]} g(t) \]
and, consequently,
\[ N[f(t),\delta; a,b]\geq N[g(t),\delta; a,b]. \]
We proceed to the proof of the theorem.
Let \(L_0\ne L_1\); then \(f_0(t)\) is not equivalent to \(\overline{f_0(t)}\). The latter is possible in the case when, on at least one \([a_i,b_i]\), \(f_0(t)\) is not equivalent to the function
\[ \frac{f_0(a_i)(b_i-t)+f_0(b_i)(t-a_i)}{b_i-a_i}, \]
that is, for \(t\in [a_i,b_i]\), \(f_0(t)\) is not monotone. The latter means that one can indicate three points \(t_1,t_2,t_3\in [a_i,b_i]\) such that \(t_1<t_2<t_3\) and \(f_0(t_1)=f_0(t_3)\ne f_0(t_2)\). We may assume that
\[ f_0(a_i)<f_0(t_1)=f_0(t_2)<f_0(b_i). \tag{19} \]
Let further, for \(t=t_4\in [a_i,b_i]\),
\[ f_0(t_4)=f_0(t_1). \]
Such a \(t_4\in [a_i,b_i]\) exists by virtue of (19).
Put
\[ f'_0(t)= \begin{cases} f_0(t), & \text{for } t\in [a,b]\setminus (a_i,b_i),\\[6pt] f_0\!\left[\dfrac{a_i(t_4-t)+t_1(t-a_i)}{t_4-a_i}\right], & \text{for } t\in [a_i,t_4],\\[10pt] f_0\!\left[\dfrac{t_4(b_i-t)+t_3(t-t_4)}{b_i-t_4}\right], & \text{for } t\in [t_4,b_i]. \end{cases} \]
It is clear that \(f'_0(t)\) is continuous on \([a,b]\), and by Lemma 3, for any \(\delta>0\),
\[ N[f'_0(t),\delta;\,a,b]\geq N[\overline{f}_0(t),\delta;\,a,b]. \tag{20} \]
By Lemma 2, for any \(\delta>0\), we have
\[ N[f'_0(t),\delta;\,a,b]\leq N[f'_0(t),\delta;\,a,t_4]+ \]
\[ +\,N[f'_0(t),\delta;\,t_4,b]+1 \tag{21} \]
and
\[ N[f_0(t),\sigma;\,a,t_1]+N[f_0(t),\delta;\,t_1,t_3]+ \]
\[ +\,N[f'_0(t),\delta;\,t_3,b]<N[f_0(t),\sigma;\,a,b]. \tag{22} \]
By Lemma 1, for any \(\delta>0\), we have
\[ N[f'_0(t),\delta;\,a,t_4]=N[f_0(t),\delta;\,a,t_1] \]
and
\[ N[f'_0(t),\delta;\,t_4,b]=N[f_0(t),\delta;\,t_3,b]. \tag{23} \]
If we put
\[ \delta\leq \sigma=|f_0(t_1)-f_0(t_2)|, \]
then we obtain
\[ N[f_0(t),\delta;\,t_1,t_3]\geq 2. \tag{24} \]
Taking (23) and (24) into account, from (22) we have
\[ N[f'_0(t),\sigma;\,a,t_4]+N[f'_0(t),\delta;\,t_4,b]+2\leq N[f_0(t),\delta;\,a,b]. \tag{25} \]
From (21) and (25), for \(\delta\leq \delta_0\) we have
\[ N[f_0(t),\delta;\,a,b]>N[f'_0(t),\delta;\,a,b]. \tag{26} \]
Further, if we put \(\delta\leq \delta_0\), then from (20) and (26) we obtain
\[ N[f_0(t),\delta;\,a,b]>N[\overline{f}_0(t),\delta;\,a,b]. \tag{27} \]
By Lemma 3 and the corollary to Lemma 1, for any \(\delta>0\) we have
\[ N[\overline{f}_{i-1}(t),\delta;\,a,b]\geq N[\overline{f}_i(t),\delta;\,a,b], \tag{28} \]
where \(\overline{f}_i(t)\in(3)\).
From (27) and (28), for \(\delta<\delta_0\) we have
\[ N[f_0(t),\delta;\,a,b]>N[\overline{f}_{n-1}(t),\delta;\,a,b]. \]
The last inequality, taking into account the corollary to Lemma 1, means that \(f_0(t)\) is not equivalent to \(f_{n-1}(t)\), that is,
\[ L_0\ne L_n. \]
The theorem proved naturally generalizes to the case of \(n\)-dimensional Euclidean space \(E_n\). We shall show how this is done in the case \(E_2\).
Let \(L_0\) be a continuous curve in the space \(E_2\).
Consider the sequences
\[ L_0,\ L_1,\ldots,L_n; \tag{29} \]
\[ \{f(t)\},\ \{f_1(t)\},\ldots,\{f_n(t)\}; \tag{30} \]
\[ f_0(t),\ \overline{f_0}(t),\ \overline{f_1}(t),\ldots,\overline{f_{n-1}}(t); \tag{31} \]
\[ G,\ G_1,\ G_2,\ldots,G_n, \tag{32} \]
analogous to the sequences (1), (2), (3), (4). If \(f_0(t)\) and \(\overline{f_0}(t)\) are not parametrizations of one and the same curve, then an interval \((a_i,b_i)\) is found such that \(f_0(t)\ne \overline{f_0}(t)\) for \(t\in(a_i,b_i)\). And take the line
\[ y=\frac{a_i-b_i}{f_0(b_i)-f_0(a_i)}\,t. \tag{34} \]
Project the sequence (31) onto the line (34). We obtain a sequence
\[ g_0(t),\ \overline{g_0}(t),\ \overline{g_1}(t),\ldots,\overline{g_{n-1}}(t) \]
of continuous mappings of the segment \([a,b]\) into the line (24). It is easy to see that
\[ N\,[g_0(t),\delta;\ a,b]\ne [\overline{g_{n-1}}(t),\varepsilon;\ a,b]. \tag{35} \]
It follows from (35) that the mappings of the segment \([a,b]\) into \(E\), \(f_0(t)\) and \(f_{n-1}(t)\), are not equivalent, that is,
\[ L\ne L_n. \]
REFERENCES
-
A. N. Kolmogorov, S. V. Fomin. Elements of the Theory of Functions and Functional Analysis. Moscow State University, 1954.
-
L. E. Portnov. Proceedings of the Second Siberian Conference. Tomsk State University Press, p. 51, 1962.