MATHEMATICS

N. V. EFIMOV and S. B. STECHKIN

CHEBYSHEV SETS IN BANACH SPACES

(Presented by Academician I. N. Vekua on 5 IV 1958)

No. 1. A set \(M\) lying in a metric space \(R\) will be called Chebyshev if for every point \(x \in R\) there exists a unique point \(y \in M\) such that the distance \(\rho(x,y)=\rho(x,M)\) (in this connection see \((^1)\)).

In this note we establish the convexity of compact Chebyshev sets in Banach spaces under certain restrictions on the unit sphere, namely, in spaces with a smooth unit sphere having uniformly small skewness (see No. 6). In particular, this thereby establishes the convexity of compact Chebyshev sets in Hilbert space.

No. 2. In what follows \(X\) denotes a real Banach space; \(E_a(p)\) denotes the open ball of radius \(a\) with center \(p \in X\); and \(O_a(p)\) denotes the complement of \(E_a(p)\) in \(X\).

Let a set \(M \subset X\) and a plane \(L \subset X\) be given. Denote by \((L,M,a)\) the intersection of \(L\) and all \(O_a(p)\), \(p \in L\), that contain \(M\). Then:

1) \((L,M,a)\) is closed (as the intersection of closed sets).

2) If \(M \subseteq M'\), then \((L,M,a) \subseteq (L,M',a)\).

3) If \(a_1 < a_2\), then \((L,M,a_1) \subseteq (L,M,a_2)\).

Property 2) follows immediately from the definition. Property 3) is also seen directly; indeed, if \(x \in L\) does not belong to \((L,M,a_2)\), i.e. lies inside some ball \(E_{a_2}(p_2)\), \(p_2 \in L\), which does not contain \(M\), then \(x\) lies inside a ball \(E_{a_1}(p_1)\), \(p_1 \in L\), which is contained in the ball \(E_{a_2}(p_2)\). Thus the complement of \((L,M,a_1)\) in \(L\) contains the complement of \((L,M,a_2)\); hence \((L,M,a_1) \subseteq (L,M,a_2)\).

No. 3. In the case when \(L \equiv X\), we shall denote \((L,M,a)\) by \((M,a)\) and call this set the \(a\)-envelope of the set \(M\). On the other hand, when \(L \ne X\), we shall often consider sets inside \(L\), abstracting from \(X\); in such cases we shall call \(L\) a space, considering that some point of \(L\) has been taken as the origin.

The following properties of \(a\)-envelopes hold:

4) \(M \subseteq (M,a)\) for every \(a\).

5) \((L,M,a_0)\) coincides with its \(a\)-envelope in the space \(L\) for every \(a \le a_0\) (proved analogously to 3), taking into account 4)).

6) The closure of the set-theoretic sum of all \((L,M,a)\), \(0<a<\infty\), coincides with its \(a\)-envelope in the space \(L\) for every \(a\) (proved analogously to 5)).

No. 4. Consider an \(n\)-dimensional Banach space \(X_n\) with a smooth unit sphere, i.e. one having at each point of the unit sphere a unique supporting hyperplane.

Lemma 1. Let \(K\) be the set of vertices of a full-dimensional simplex \(S \subset X_n\); then the closure of the set-theoretic sum of all \(a\)-envelopes of \(K\) is the simplex \(S\).

Lemma 2. Let \(M \subset X_n\); if \(M \equiv (M,a)\) for every \(a\), then \(M\) is either convex or degenerate (i.e. lies in some hyperplane).

Lemma 2 follows from Lemma 1.

No. 5. Here we again consider an infinite-dimensional Banach space \(X\). We call a closed ball \(\bar E\) a supporting ball to a certain set \(Q\) at the points of its subset \(Q_1\), if inside \(\bar E\) there are no points of \(Q\), and \(Q_1\) lies on the boundary of \(\bar E\).

Lemma 3. If \(M \subset X\) is compact, then for any boundary point \(y_0\) of the set \((L,M,a)\), regarded as a set of the space \(L\), there exists a closed ball \(\bar E_a(p)\), \(p \in L\), supporting \((L,M,a)\) at the point \(y_0\) and also supporting \(M\).

Lemma 4 (on removing a ball). Let \(M \subset X\) be a compact set; \(\bar E\) a closed ball supporting \(M\) at the single point \(x_0\); \(e\) a vector going from \(x_0\) into the interior of the ball \(\bar E\). Then there exists \(\lambda_0>0\) such that the ball

\[ \bar E'=\bar E+\lambda e \]

does not intersect \(M\) for all \(\lambda\), \(0<\lambda\le \lambda_0\).

Lemma 5. Let the ball \(\bar E\) be smooth; \(A\) a supporting hyperplane to \(\bar E\) at an arbitrary boundary point \(x\); \(e\) a vector which, when applied at \(x\), goes into that (open) half-space of \(X\), relative to \(A\), in which the ball \(\bar E\) lies. Then \(e\) goes into the interior of the ball \(\bar E\).

Proof. Suppose that no point of the segment \(z=x+te\), \(0\le t\le t_0\), is an interior point of \(\bar E\). Denote by \(W\) the convex hull of this segment and \(\bar E\). The closed set \(W\) is a convex body; \(z=x+te\), \(0<t<t_0\), is its boundary point. By Mazur’s theorem \((^2)\), at the point \(z\) there exists a hyperplane \(B\) supporting \(W\). This hyperplane contains the entire segment \(z=x+te\), \(0\le t\le t_0\), and, consequently, is a supporting hyperplane to the ball \(\bar E\) at the point \(x_0\). Moreover, by construction, \(B\) does not coincide with \(A\). We have arrived at a contradiction with the definition of smoothness of the ball \(\bar E\). Hence it follows that there exists a point \(z_1=x+t_1e\), \(0<t_1\le t_0\), interior to \(\bar E\); consequently, the whole segment \(z=x+te\), \(0<t\le t_1\), consists of interior points of \(\bar E\). The lemma is proved.

No. 6. Let \(X\) be a strictly convex Banach space; \(S\) its unit sphere; \(L_n\) an \(n\)-dimensional subspace; \(x\in S\) an arbitrary point whose distance from \(L_n\) is \(\le \varepsilon<1\); \(\Lambda_{n-1}\) an \((n-1)\)-dimensional plane tangent to \(S\) at the point \(x\) and parallel to \(L_n\). Consider the intersection of \(S\) and \(L_n\); owing to the strict convexity of \(X\) there will be exactly two points \(y_1,y_2\in L_n\) at which the \((n-1)\)-dimensional tangent plane to \(S\) is parallel to \(\Lambda_{n-1}\); denote by \(y\) the one of them nearest to the point \(x\); by \(\rho(x,y)\) the distance between \(x\) and \(y\). We shall say that \(X\) is a space with uniformly small skewness if there exists a number \(k\) \((k>1)\) such that

\[ \rho(x,y)\le f(\varepsilon),\qquad f(\varepsilon)=k\varepsilon, \]

for all \(x\in S\).

A space with uniformly small skewness has the following property, which we shall use below: if \(L_n\) is an \(n\)-dimensional plane passing through the center \(p\) of the sphere \(S_a(p)\), and all the other notations retain the analogous meaning, then \(\rho(x,y)\le f(\varepsilon)\) for all \(a\) and \(x\in S_a(p)\).

Remark. It can be shown that not every uniformly convex space is a space with uniformly small skewness; however, Hilbert space belongs to this type (for it \(\rho(x,y)\le \varepsilon\sqrt{2}\)).

No. 7. Let \(M\) be a Chebyshev compact set in a space with uniformly small skewness and a smooth unit sphere. For any \(\varepsilon>0\), choose some \(\varepsilon\)-net of the set \(M\), construct the smallest finite-dimensional plane \(L_\varepsilon\) passing through all points of this \(\varepsilon\)-net, and construct the set \((L_\varepsilon,M,a)\).

Lemma 6. For any \(a>0\), every point of the set \((L_\varepsilon,M,a)\) is at distance \(\leq f(\varepsilon)\) from \(M\).

Proof. From the definition of the set \((L_\varepsilon,M,a)\) it follows that it is bounded and, consequently, has boundary points in \(L_\varepsilon\). Consider any one of its boundary points \(y_0\). According to Lemma 3 there exists a closed ball \(\overline E_a(p)\), supporting the set \((L_\varepsilon,M,a)\) at the point \(y_0\) and at the same time supporting \(M\). Since \(M\) is a Chebyshev set, \(\overline E_a(p)\) intersects \(M\) in the single point \(x_0\). We shall prove that \(\rho(x_0,y_0)\leq f(\varepsilon)\). Denote by \(A\) and \(B\) the hyperplanes in \(X\) supporting \(\overline E_a(p)\) respectively at the points \(x_0\) and \(y_0\). If \(\rho(x_0,y_0)>f(\varepsilon)\), then the intersections \(A\) and \(B\) with \(L_\varepsilon\) are not parallel. Therefore there is a vector \(e\) which belongs to \(L_\varepsilon\) and satisfies the following two conditions: 1) being applied to some point of the hyperplane \(A\), it is directed into that open half-space of \(X\) with respect to \(A\) in which the ball \(\overline E_a(p)\) lies; 2) being applied to some point of the hyperplane \(B\), it is directed into that open half-space of \(X\) with respect to \(B\) in which the ball \(\overline E_a(p)\) does not lie. By Lemma 5 and the first of these two conditions, the vector \(e\), with point of application \(x_0\), is directed into the interior of the ball \(\overline E_a(p)\). On the basis of Lemma 4 there exists a number \(\lambda_0>0\) such that the ball

\[ \overline E_a(p')=\overline E_a(p)+\lambda e,\qquad 0<\lambda\leq \lambda_0, \]

does not intersect \(M\), and moreover has its center \(p'\) in the plane \(L_\varepsilon\). By virtue of the second condition in the definition of the vector \(e\), for sufficiently small \(\lambda\) the point \(y_0\) will lie inside the ball \(\overline E_a(p')\), i.e. it will be an exterior point of the set \((L_\varepsilon,M,a)\) in the space \(L_\varepsilon\).

We have obtained a contradiction to the condition that \(y_0\) is a boundary point of \((L_\varepsilon,M,a)\). Thus the inequality \(\rho(y_0,x_0)\leq f(\varepsilon)\) has been proved by contradiction. Consequently, \(\rho(y_0,M)\leq f(\varepsilon)\).

Now we shall prove that every interior point of the set \((L_\varepsilon,M,a)\) in the space \(L_\varepsilon\) is likewise at distance \(\leq f(\varepsilon)\) from \(M\).

Suppose that inside \((L_\varepsilon,M,a)\) in the space \(L_\varepsilon\) there is a point \(y\) for which \(\rho(y,M)>f(\varepsilon)\). Since the set \((L_\varepsilon,M,a)\) is compact, the continuous function \(\rho(y,M)\) attains on it a maximum \(m=\rho(q,M)>f(\varepsilon)\), \(q\in L_\varepsilon\). The point \(q\) lies inside \((L_\varepsilon,M,a)\), since, according to what has been proved, on the boundary we have \(\rho(x,M)\leq f(\varepsilon)\). Consider in \(X\) the ball \(\overline E_m(q)\). This ball has a unique point of intersection with \(M\), namely \(x_0\). Since \(\varepsilon\leq f(\varepsilon)<m\), the supporting hyperplane \(C\) of the ball \(\overline E_m(q)\) at the point \(x_0\) is not parallel to \(L_\varepsilon\). Therefore there exists a vector \(e\in L_\varepsilon\) which, being applied to the point \(x_0\), is directed into that half of the space \(X\) with respect to the hyperplane \(C\) in which the ball \(\overline E_m(q)\) lies, i.e. into the interior of this ball. Again applying Lemma 4, we find that there exists \(\lambda_0>0\) such that the ball

\[ \overline E_m(q_1)=\overline E_m(q)+\lambda e,\qquad 0<\lambda\leq \lambda_0, \]

has no common points with \(M\). For sufficiently small \(\lambda\), the point \(q_1\) will lie inside \((L_\varepsilon,M,a)\) in the space \(L_\varepsilon\), and moreover \(\rho(q_1,M)>m\).

We have arrived at a contradiction to the definition of the number \(m\). Thus it has been proved that for every point \(y\in(L_\varepsilon,M,a)\) the relation \(\rho(y,M)\leq f(\varepsilon)\) holds, i.e. the lemma is proved.

Theorem. A Chebyshev compact set in a Banach space with uniformly small skewness and a smooth unit sphere is a convex set.

Proof. Let \(G_\varepsilon\) be the closure of the set-theoretic sum of all \((L_\varepsilon, M, a)\), \(0<a<\infty\). According to No. 4,

\[ G_\varepsilon=(G_\varepsilon,a)\ \text{in}\ L_\varepsilon . \]

By the construction of \(L_\varepsilon\), the set \(G_\varepsilon\) is nondegenerate in the space \(L_\varepsilon\). Hence, by Lemma 2, \(G_\varepsilon\) is convex. By Lemma 6, the distance from any point \(y\in G_\varepsilon\) to the set \(M\) satisfies the inequality

\[ \rho(y,M)\leq f(\varepsilon). \]

By the definition of the function \(f(\varepsilon)\), we have \(f(\varepsilon)\to0\) as \(\varepsilon\to0\).

Thus, the compact set \(M\) is approximated arbitrarily well by convex sets. It follows that \(M\) is convex. The theorem is proved.

Moscow State University
named after M. V. Lomonosov

Received
4 IV 1958

References

¹ N. V. Efimov, S. B. Stechkin, DAN, 118, No. 1, 17 (1958). ² S. Mazur, Stud. Math., 4, 70 (1933).