Abstract
Authors
A.B. Nemeth
Institutul de Calcul
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A.B. Németh, About an imbedding conjecture for k-independent sets. Fund. Math. 67 1970 203–207.
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Fundamenta Mathematicae
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Institute of Mathematics Polish Academy of Sciences
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0016-2736
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1730-6329
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About an imbedding conjecture for kk-independent sets
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Following [1] we say that a subset XX of the nn-dimensional real Euclidean space R^(n)R^{n} is kk-independent ( 0 <= k <= n-10 \leqslant k \leqslant n-1 ) if any k+2k+2 distinct points of that subset are linearly independent. ( ^(1){ }^{1} )
In what follows the homeomorphic image of the set {(x^(1),dots,x^(m)):}\left\{\left(x^{1}, \ldots, x^{m}\right)\right. : {: sum(x^(i))^(2) < 1}\left.\sum\left(x^{i}\right)^{2}<1\right\} in R^(m)R^{m} will be said to be an open mm-cell; the homeomorphic image of the set {(x^(1),dots,x^(m)):sum(x^(i))^(2)=1}\left\{\left(x^{1}, \ldots, x^{m}\right): \sum\left(x^{i}\right)^{2}=1\right\} will be said to be an m-1m-1-sphere.
K. Borsuk [1] has proved the following imbedding theorem concerning kk-independent sets:
K. Borsuk [1] has proved the following imbedding theorem concerning kk-independent sets:
If XX is a compact kk-independent set in R^(n)R^{n} and if NN is an open subset in XX containing kk distinct points, then X\\NX \backslash N is homeomorphic with a subset of R^(n-k)R^{n-k}.
In [6], p. 503 and in [4], another notion of kk-independence is applied, which is useful in applications in the approximation theory and which will be called in the sequel kk-vectorial-independence.
The subset XX of R^(n)R^{n} will be said to be kk-vectorial-independent if for any kk of its distinct points x_(1),dots,x_(k)x_{1}, \ldots, x_{k} the vectors vec(Ox_(1)),dots, vec(Ox_(k))\overrightarrow{O x_{1}}, \ldots, \overrightarrow{O x_{k}}, where OO is the origin in R^(n)R^{n}, are linearly independent.
Observation 1. A kk-vectorial-independent subset XX in R^(n)R^{n} is k-2k-2-independent in the sense of [1].
Indeed, if x_(1),dots,x_(k)x_{1}, \ldots, x_{k} are kk distinct points in XX, then they cannot be contained in any k-2k-2-dimensional hyperplane H^(k-2)H^{k-2}, because such a hyperplane generates a k-1k-1-dimensional subspace (i.e. a k-1k-1-dimensional hyperplane passing through the origin), and if x_(1),dots,x_(k)x_{1}, \ldots, x_{k} were in H^(k-2)H^{k-2}, the vectors vec(Ox_(1)),dots, vec(Ox_(k))\overrightarrow{O x_{1}}, \ldots, \overrightarrow{O x_{k}} would be linearly dependent, being in R^(k-1)R^{k-1}.
Observation 2. If XX is a kk-independent subset in R^(n)R^{n}, then it may be considered a k+2k+2-vectorial-independent subset in R^(n+1)R^{n+1} if we consider R^(n)R^{n} as a hyperplane H^(n)H^{n} in R^(n+1)R^{n+1} not passing through the origin.
Indeed, consider k+2k+2 distinct points x_(1),dots,x_(k+2)x_{1}, \ldots, x_{k+2} in XX in H^(n)H^{n}. The vectors vec(Ox_(1)),dots, vec(Ox_(k+2))\overrightarrow{O x_{1}}, \ldots, \overrightarrow{O x_{k+2}} in R^(n+1)R^{n+1} are linearly independent. If they were linearly dependent, there would exist a subspace R^(k+1)R^{k+1} in R^(n+1)R^{n+1} containing them, which would intersect H^(n)H^{n} in a kk-dimensional hyperplane H^(k)H^{k} containing the points x_(1),dots,x_(k+2)x_{1}, \ldots, x_{k+2}, which is a contradiction.
It was conjectured by A. M. Gleason (see [8]) that the kk-independent compact subset XX in R^(n)R^{n} is homeomorphic with a subset of S^(n-k)S^{n-k}, the n-kn-k-sphere. Investigations about this imbedding conjecture were announced by C. T. Yang in [8], but we have not been able to obtain any information about his results.
In Theorem 2 of [4] it was proved that if XX is a compact kk-vectorialindependent set in R^(n),NR^{n}, N is open in XX and contains k-2k-2 distinct points, then X\\NX \backslash N is homeomorphic with a subset of R^(n-k+1)R^{n-k+1}, which is an analogue of the imbedding theorem of KK. Borsuk for kk-vectorial-independent sets. By a similar reformulation of the conjecture of A. M. Gleason, we obtain:
If XX is a kk-vectorial-independent compact subset of R^(n)R^{n}, then it is homeomorphic with a subset of S^(n-k+1)S^{n-k+1}.
Making use of our Observation 2 above, we can see that this conjecture implies the conjecture of Gleason. Our conjecture for k=nk=n is the well-known theorem of J. Mairhuber [3] in the approximation theory. For k=1k=1 it is obviously true, and for k=2k=2 an imbedding of XX into a proper subset of S^(n-1)S^{n-1} may be realised by the radial projection with respect to the origin of R^(n)R^{n} into the geometrical sphere with its centre at the origin.
Let XX be a compact kk-independent set in R^(n)R^{n} containing an mm-cell. Then, as has been proved by S. S. Ryškov [5], the following inequality is valid: (^(2))\left({ }^{2}\right)
[(k+2)/(2)]m+[(k+1)/(2)] <= n.\left[\frac{k+2}{2}\right] m+\left[\frac{k+1}{2}\right] \leqslant n .
Suppose now that the compact kk-vectorial-independent subset XX in R^(n)R^{n} is of dimension mm. Then there exists a closed subset X_(0)X_{0} of XX of the same dimension mm and an n-1n-1-hyperplane H^(n-1)H^{n-1} which has the property of separating strictly the origin OO and the subset X_(0)X_{0}. Denote by X_(0)^(')X_{0}^{\prime} the radial projection with respect to 0 of X_(0)X_{0} into H^(n-1)H^{n-1}. Obviously, X_(0)^(')X_{0}^{\prime} is a kk-vectorial-independent homeomorphic image of X_(0)X_{0}, and therefore
(^(2))\left({ }^{2}\right) In [5], Ryškov defines the so-called kk-regular sets as being in fact kk-independent in the sense of [1], and has announced his inequality for these sets. But all the reasonings in the text are valid for kk-l-independent sets. In Uspehi Mat. Nauk 15 (6) (1960), pp. 125-132, the definition of the kk-regular sets is changed in this sense. Our inequality follows from the inequality of Ryskov applied to kk-independent sets.
from Observation 1 it follows that X_(0)^(')X_{0}^{\prime} is a k-2k-2-independent subset of H^(n-1)H^{n-1}. Applying the inequality of S. S. Ryškov we conclude that
(^(2))\left({ }^{2}\right) In [5], Ryškov defines the so-called kk-regular sets as being in fact kk-independent in the sense of [1], and has announced his inequality for these sets. But all the reasonings in the text are valid for kk-l-independent sets. In Uspehi Mat. Nauk 15 (6) (1960), pp. 125-132, the definition of the kk-regular sets is changed in this sense. Our inequality follows from the inequality of Ryskov applied to kk-independent sets.
from Observation 1 it follows that X_(0)^(')X_{0}^{\prime} is a k-2k-2-independent subset of H^(n-1)H^{n-1}. Applying the inequality of S. S. Ryškov we conclude that
[(k)/(2)]m+[(k-1)/(2)] <= n-1\left[\frac{k}{2}\right] m+\left[\frac{k-1}{2}\right] \leqslant n-1
The present note aims at giving a proof of our conjecture in the particular case where the kk-vectorial-independent set XX in R^(n)R^{n} contains an n-k+1n-k+1-cell. More precisely, we shall prove the following.
THEOREM. Let XX be a compact subset of R^(n),n >= 2R^{n}, n \geqslant 2, which is kk-vectorialindependent and contains an n-k+1n-k+1-cell. Then XX is homeomorphic with a subset of S^(n-k+1)S^{n-k+1}.
The inequality (*) restricts kk in this case to k <= 3k \leqslant 3 or k=nk=n. As we have observed above, in the case of k=2k=2 the proof is simple, and in the case of k=nk=n it is known. Therefore only the case k=3k=3 will be considered, and to justify our theorem in this case, we observe that each geometrical sphere S^(n-2)S^{n-2} in a hyperplane H^(n-1)H^{n-1} in R^(n)R^{n} not passing through the origin is a 3 -vectorial-independent set in R^(n)R^{n} containing n-2n-2-cells.
In the proof given here we apply a method utilised by I. J. Schoenberg and C. T. Yang in [7] for proving the theorem of J. Mairhuber. An important moment in the proof is the employing of the following theorem of M. Brown [2]:
If hh is a homeomorphic imbedding of S^(n-1)xx IS^{n-1} \times I into S^(n)S^{n}, then the closure of either complementary domain of h(S^(n-1)xx{1//2})h\left(\mathbb{S}^{n-1} \times\{1 / 2\}\right) in S^(n)\mathbb{S}^{n} is a closed nn-cell. (Here I=[0,1]I=[0,1].)
We begin with a lemma:
Lemma. Let XX be a compact Hausdorff space having the following properties:
(i) XX contains an open nn-cell QQ as an open subset;
(ii) if NN is a non-empty open subset of XX, then X\\NX \backslash N may be imbedded in a proper subset of S^(n)S^{n}.
Lemma. Let XX be a compact Hausdorff space having the following properties:
(i) XX contains an open nn-cell QQ as an open subset;
(ii) if NN is a non-empty open subset of XX, then X\\NX \backslash N may be imbedded in a proper subset of S^(n)S^{n}.
Then XX is homeomorphic with a subset of S^(n)S^{n}.
Proof. If XX is not connected, the proof is immediate.
Suppose that XX is connected. Let AA be an annulus, that is to say the homeomorphic image of the set S^(n-1)xx IS^{n-1} \times I, which is contained in the nn-cell QQ. Since QQ is open in X,AX, A separates XX and so does sigma^(n-1)\sigma^{n-1}, the image in AA of S^(n-1)xx{1//2}S^{n-1} \times\{1 / 2\}, i.e. X\\A=Y_(1)uuY_(2),X\\sigma^(n-1)=V_(1)uuV_(2)X \backslash A=Y_{1} \cup Y_{2}, X \backslash \sigma^{n-1}=V_{1} \cup V_{2}, where Y_(1),Y_(2)Y_{1}, Y_{2}, and respectively V_(1),V_(2)V_{1}, V_{2} are non-empty open disjoint subsets of XX. Suppose that Y_(1)subV_(1),Y_(2)subV_(2)Y_{1} \subset V_{1}, Y_{2} \subset V_{2} and introduce the notations: B_(1)=V_(1)uusigma^(n-1),B_(2)=V_(2)uusigma^(n-1)B_{1}=V_{1} \cup \sigma^{n-1}, B_{2}=V_{2} \cup \sigma^{n-1}. The sets B_(1)B_{1} and B_(2)B_{2} are both 'connected and are not separated by sigma^(n-1)\sigma^{n-1}. Denote by ff and by gg the homeomorphisms of X\\Y_(2)X \backslash Y_{2} and, respectively, of X\\Y_(1)X \backslash Y_{1} in S^(n)S^{n}, which exist according to (ii). Since the above sets both contain AA, from the theorem of M . Brown [2]
it follows that the complementary domains of fsigma^(n-1)f \sigma^{n-1} and gsigma^(n-1)g \sigma^{n-1} in S^(n)S^{n} are open nn-cells. Thus we may suppose that ff and gg are homeomorphisms which both transform sigma^(n-1)\sigma^{n-1} in the equator EE of S^(n)S^{n} and by which B_(1)B_{1} is mapped into the north hemisphere and B_(2)B_{2} into the south hemisphere of S^(n)S^{n} (suppose that S^(n)S^{n} is a geometrical sphere). Consider the following homeomorphism of EE onto itself: l=g@f^(-1)∣El=g \circ f^{-1} \mid E. Let hh be an extension of the homeomorphism ll to a homeomorphism of the whole north hemisphere onto itself. Then h@fh \circ f will be a homeomorphism of B_(1)B_{1} into the north hemisphere carrying sigma^(n-1)\sigma^{n-1} onto EE. Consider the mapping
Proof. If XX is not connected, the proof is immediate.
Suppose that XX is connected. Let AA be an annulus, that is to say the homeomorphic image of the set S^(n-1)xx IS^{n-1} \times I, which is contained in the nn-cell QQ. Since QQ is open in X,AX, A separates XX and so does sigma^(n-1)\sigma^{n-1}, the image in AA of S^(n-1)xx{1//2}S^{n-1} \times\{1 / 2\}, i.e. X\\A=Y_(1)uuY_(2),X\\sigma^(n-1)=V_(1)uuV_(2)X \backslash A=Y_{1} \cup Y_{2}, X \backslash \sigma^{n-1}=V_{1} \cup V_{2}, where Y_(1),Y_(2)Y_{1}, Y_{2}, and respectively V_(1),V_(2)V_{1}, V_{2} are non-empty open disjoint subsets of XX. Suppose that Y_(1)subV_(1),Y_(2)subV_(2)Y_{1} \subset V_{1}, Y_{2} \subset V_{2} and introduce the notations: B_(1)=V_(1)uusigma^(n-1),B_(2)=V_(2)uusigma^(n-1)B_{1}=V_{1} \cup \sigma^{n-1}, B_{2}=V_{2} \cup \sigma^{n-1}. The sets B_(1)B_{1} and B_(2)B_{2} are both 'connected and are not separated by sigma^(n-1)\sigma^{n-1}. Denote by ff and by gg the homeomorphisms of X\\Y_(2)X \backslash Y_{2} and, respectively, of X\\Y_(1)X \backslash Y_{1} in S^(n)S^{n}, which exist according to (ii). Since the above sets both contain AA, from the theorem of M . Brown [2]
it follows that the complementary domains of fsigma^(n-1)f \sigma^{n-1} and gsigma^(n-1)g \sigma^{n-1} in S^(n)S^{n} are open nn-cells. Thus we may suppose that ff and gg are homeomorphisms which both transform sigma^(n-1)\sigma^{n-1} in the equator EE of S^(n)S^{n} and by which B_(1)B_{1} is mapped into the north hemisphere and B_(2)B_{2} into the south hemisphere of S^(n)S^{n} (suppose that S^(n)S^{n} is a geometrical sphere). Consider the following homeomorphism of EE onto itself: l=g@f^(-1)∣El=g \circ f^{-1} \mid E. Let hh be an extension of the homeomorphism ll to a homeomorphism of the whole north hemisphere onto itself. Then h@fh \circ f will be a homeomorphism of B_(1)B_{1} into the north hemisphere carrying sigma^(n-1)\sigma^{n-1} onto EE. Consider the mapping
varphi x={[h@fx," for ",x" in "B_(1)","],[gx," for ",x" in "B_(2).]:}\varphi x=\left\{\begin{array}{lll}
h \circ f x & \text { for } & x \text { in } B_{1}, \\
g x & \text { for } & x \text { in } B_{2} .
\end{array}\right.
varphi\varphi is a well-defined mapping which is one-to-one and continuous. To prove its continuity, let U sub varphi XU \subset \varphi X be open. If U nn E=O/U \cap E=\emptyset, then varphi^(-1)U\varphi^{-1} U is open according to the continuity of h@fh \circ f and gg. Suppose U nn E!=O/U \cap E \neq \varnothing. Then the sets h@fB_(1)nn Uh \circ f B_{1} \cap U and gB_(2)nn Ug B_{2} \cap U are open in the relative topology of h@fB_(1)h \circ f B_{1} and gB_(2)g B_{2} respectively. Therefore the sets
W_(1)=varphi^(-1)(h@fB_(1)nn U)=(h@f)^(-1)(h@fB_(1)nn U)W_{1}=\varphi^{-1}\left(h \circ f B_{1} \cap U\right)=(h \circ f)^{-1}\left(h \circ f B_{1} \cap U\right)
and
W_(2)=varphi^(-1)(gB_(2)nn U)=g^(-1)(gB_(2)nn U)W_{2}=\varphi^{-1}\left(g B_{2} \cap U\right)=g^{-1}\left(g B_{2} \cap U\right)
are open in the relative topology of B_(1)B_{1} and B_(2)B_{2}, respectively. Let G_(1)G_{1} and G_(2)G_{2} be open sets in XX such that G_(1)nnB_(1)=W_(1),G_(2)nnB_(2)=W_(2)G_{1} \cap B_{1}=W_{1}, G_{2} \cap B_{2}=W_{2}. Then the sets G_(1)uuV_(2)G_{1} \cup V_{2} and G_(2)uuV_(1)G_{2} \cup V_{1} are open in XX and
W_(1)uuW_(2)=(G_(1)uuV_(2))nn(G_(2)uuV_(1)).W_{1} \cup W_{2}=\left(G_{1} \cup V_{2}\right) \cap\left(G_{2} \cup V_{1}\right) .
But W_(1)uuW_(2)=varphi^(-1)UW_{1} \cup W_{2}=\varphi^{-1} U, which completes the proof of the continuity of varphi\varphi. From the compactness of XX it follows that varphi\varphi is a homeomorphic imbedding of XX into S^(n)S^{n}.
Proof of the theorem. If XX in the theorem contains an n-2n-2-cell (remember that only the case k=3k=3 is considered), then it contains an open n-2n-2 - cell as an open set. Indeed, suppose that QQ is an open n-2n-2-cell in XX such that X\\ bar(Q)!=O/X \backslash \bar{Q} \neq \emptyset. According to Theorem 2 in [4], QQ is open in any closed proper subset in XX in which it is contained. Then QQ is open in XX according to the normality of this space. From Theorem 2 in [4] it also follows that X\\NX \backslash N may be topologically imbedded into a proper subset of S^(n-2)S^{n-2} for any non-empty, open subset NN in XX. It follows that all the conditions of the lemma are satisfied and therefore XX is homeomorphic with a subset of S^(n-2)S^{n-2}.
References
[1] K. Borsuk, On the kk-independent sets of the Euclidean space and of the Hilbert space, Bull. Acad. Sci. Pol. 5 (1957), pp. 351-356.
[2] M. Brown, A proof of the generalised Schoenflies theorem, Bull. Amer. Math. Soc. 66 (1960), pp. 74-76.
[3] J. C. Mairhuber, On Haar's theorem concerning Chebyshev approximation problems having unique solutions, Proc. Amer. Math. Soc. 7 (1956), pp. 609-615.
[4] A. B. Németh, Homeomorphe projections of kk-independent sets and Chebyshev subspaces of finite dimensional Chebyshev spaces, Mathematica 9 (1967), pp. 325-333.
[5] S. S. Ryškov (С. С. Рышков), О к-регулярных вложениях, Докл. Акад. Наук. СССР 127 (1959), pp. 272-273.
[6] Yu. A. Šaškin (Ю. А. Шашкин), Систем Коровкина в пространствах непрерывных функций, Изв. Акад. Наук. СССР 26 (1962), рр. 495-512.
[7] I. J. Schoenberg and C. T. Yang, On the unicity of solutions of problems of best approximation, Ann. Math. Pura Appl. 54, pp. 1-12.
[8] C. T. Yang, On non-orientable closed surfaces in Euclidean spaces, Canad, J. Math. 14 (1962), pp. 660-668.
[2] M. Brown, A proof of the generalised Schoenflies theorem, Bull. Amer. Math. Soc. 66 (1960), pp. 74-76.
[3] J. C. Mairhuber, On Haar's theorem concerning Chebyshev approximation problems having unique solutions, Proc. Amer. Math. Soc. 7 (1956), pp. 609-615.
[4] A. B. Németh, Homeomorphe projections of kk-independent sets and Chebyshev subspaces of finite dimensional Chebyshev spaces, Mathematica 9 (1967), pp. 325-333.
[5] S. S. Ryškov (С. С. Рышков), О к-регулярных вложениях, Докл. Акад. Наук. СССР 127 (1959), pp. 272-273.
[6] Yu. A. Šaškin (Ю. А. Шашкин), Систем Коровкина в пространствах непрерывных функций, Изв. Акад. Наук. СССР 26 (1962), рр. 495-512.
[7] I. J. Schoenberg and C. T. Yang, On the unicity of solutions of problems of best approximation, Ann. Math. Pura Appl. 54, pp. 1-12.
[8] C. T. Yang, On non-orientable closed surfaces in Euclidean spaces, Canad, J. Math. 14 (1962), pp. 660-668.
THE COMPUTING INSTITUTE OF THE ACADEMY OF ROUMANIA Cluj
Reçu par la Rédaction le 5. 5. 1968
- ( ^(1){ }^{1} ) For the sake of simplicity, the affine space and the vectorial Euclidean space of dimension nn are denoted by the same symbol R^(n)R^{n}.