MATHEMATICS

FLORIN CONSTANTINESCU

ON CHEBYSHEV SETS

(Presented by Academician S. L. Sobolev, 1 IX 1959)

1. Let \(X_n\) be an \(n\)-dimensional Banach space, and let \(M\) be some set in \(X_n\). As usual, we shall call the distance from a point \(x \in X_n\) to the set \(M\) the number
   \[ \rho(x,M)=\inf_{y\in M}\|x-y\|. \]

Together with N. V. Efimov and S. B. Stechkin \((^1)\), we shall call the set \(M\): 1) a set of existence, if for every \(x \in X_n\) there is a nearest element \(y_0 \in M\) such that \(\rho(x,y)=\rho(x,y_0)\); 2) a Chebyshev set, if for every \(x \in X_n\) there is a unique element \(y_0 \in M\) such that \(\rho(x,M)=\rho(x,y_0)\).

Obviously, every closed bounded set is a set of existence, and every set of existence (in particular, every Chebyshev set) is a closed set.

N. V. Efimov and S. B. Stechkin proved the following theorem:

In an \(n\)-dimensional Banach space \(X_n\), the class of bounded Chebyshev sets coincides with the class of bounded closed convex sets if and only if the unit sphere of \(X_n\) is strictly convex and has no conical points.

1. We wish to prove that the theorem remains valid if the boundedness requirement is omitted from it.

Let \(M\) be such a set. We shall first prove that \(M\) is a set of existence.

For this purpose consider the sphere \(S(x,r+a)\) with center \(x\) and radius \(r+a\), where \(r=\rho(x,M)\) and \(a>0\). Then the set \(P_a=M\cap S(x,r+a)\) is nonempty, bounded, and closed. Hence it follows that there exists an element \(y_0\in P_a\) for which we have \(\rho(x,P_a)=\rho(x,y_0)\). On the other hand, whatever element \(y\in M\) lying outside the sphere \(S(x,r+a)\) may be, we have \(\rho(x,y)>r+a\). Consequently, \(\rho(x,M)=\rho(x,P_a)\), i.e. there exists an element \(y_0\in M\) such that \(\rho(x,M)=\rho(x,y_0)\).

Theorem 1. If in an \(n\)-dimensional Banach space \(X_n\) the unit sphere has no conical points, then every Chebyshev set is convex.

Let \(M\) be a Chebyshev set, \(x\notin M\). Then there exists a unique element for which \(\rho(x,M)=\rho(x,y_0)\). Consider the spheres with center at \(x\), whose radii are the numbers \(\rho(x,M)+1,\rho(x,M)+2,\ldots,\rho(x,M)+n,\ldots\). Forming the intersections of these spheres with the set \(M\), we obtain nonempty sets \(P_1,P_2,\ldots,P_n,\ldots\).

The sets \(P_n\) are Chebyshev sets. Indeed, since the spheres are bounded, closed, and convex sets, on the basis of Theorem 2 of N. V. Efimov and S. B. Stechkin it follows that these spheres are Chebyshev sets. Taking into account that the intersection of two Chebyshev sets is also a Chebyshev set and applying the aforementioned theorem of Efimov and Stechkin, we obtain that the sets \(P_n\) are convex. On the other hand, \(P_1\subset P_2\subset \cdots\)

… \(\subset P_n \subset \ldots\) and \(M=\bigcup_{n=1}^{\infty} P_n\); hence it follows at once that the set \(M\) is also convex. The theorem is proved.

1. If the sphere \(X_n\) is strictly convex, then it is not difficult to prove that every convex set is Chebyshev.

It is easy to see that in a space \(X_n\) in which the unit sphere is not strictly convex, there exist convex non-Chebyshev sets; and also that in a space \(X_n\) in which the unit sphere contains conical points, there exist Chebyshev sets that are not convex.

Hence, and on the basis of Theorem 1, we obtain Theorem 2:

Theorem 2. In an \(n\)-dimensional Banach space \(X_n\), the class of Chebyshev sets coincides with the class of convex sets if and only if the unit sphere is strictly convex and has no conical points.

State University
named after V. Babeș
Cluj, Romania

Received
28 VIII 1959

REFERENCES

1. N. V. Efimov, S. B. Stechkin, DAN, 118, No. 1 (1958).