MATHEMATICS

N. V. EFIMOV and S. B. STECHKIN

SOME PROPERTIES OF CHEBYSHEV SETS

(Presented by Academician S. L. Sobolev on 25 VI 1957)

1°. We consider a Banach space (X) and a certain set (M \subset X); as usual, we call the number

[
\rho(x, M)=\inf_{y\in M}|x-y|
]

the distance from (x \in X) to (M).

In connection with the problem of best approximations, we shall call (M): 1) a set of existence, if for every (x \in X) there is a nearest element (y \in M), i.e., there is a (y \in M) such that (\rho(x,M)=\rho(x,y)); 2) a set of uniqueness, if for every (x \in X) there is at most one nearest element (y \in M). A set which is simultaneously a set of existence and of uniqueness will be called Chebyshev. In what follows (M) is assumed to be bounded.

2°. The present note is devoted mainly to the following question: what are the relations between the class of Chebyshev sets and the class of convex sets? In particular, under what conditions on the Banach space (X) can one assert that every Chebyshev set (M \subset X) is convex? Below, a solution of this problem is given in the case of the (n)-dimensional space (X_n); at the same time, some simple geometric propositions are reported that are valid in an arbitrary (X).

3°. Obviously, every set of existence, and consequently every Chebyshev set, is closed. Therefore in what follows only closed sets are considered.

The simplest examples show that Banach spaces (X) (or (X_n)) are possible in which there exist Chebyshev nonconvex sets and convex non-Chebyshev sets. Indeed, let (X_2) be the Euclidean plane on which a new metric is introduced so that the Euclidean square (K) is taken as the unit circle. Then in (X_2) there exist convex sets which are not Chebyshev sets, for example (K); it is clear that the essence of the matter here lies in the presence of rectilinear segments on the unit circle. On the other hand, in (X_2) there exist Chebyshev sets which are not convex. Such a set is obtained, for example, if from a Euclidean disk we cut out a sector with an obtuse angle whose Euclidean bisector is parallel to a diagonal of the square (K); here the essence of the matter lies in the presence of angular points on the unit circle.

4°. Let (E) be the unit sphere of the space (X). The sphere is called strictly convex if from the conditions (x_0 \in X), (x_1 \in X), (x_0 \ne x_1), (|x_0|=1), (|x_1|=1) it follows that (|x_t|<1), where (x_t=(1-t)x_0+tx_1) ((0<t<1)), i.e., if (E) contains no segment.

The sphere (E) does not contain conical points (in the two-dimensional case, angular points) if for every (x_0 \in E) there exists one and only one linear functional (f_0) satisfying the conditions (|f_0|=1), (f_0(x_0)=1).

These two properties are, in a certain sense, reciprocal. Namely, let (E^) be the unit sphere of the conjugate space (X^). Then, if (E^) is strictly convex, (E) has no conical points; if (E^) has no conical points, then (E) is strictly convex.

5°. Generalizing the considerations given in 3°, it is easy to see that in any space (X) in which the sphere (E) is not strictly convex, there exist convex non-Chebyshev sets; in any space (X) in which the sphere (E) contains conical points, there exist Chebyshev but nonconvex sets. Along with this the following theorem is true:

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

If the sphere (E) is strictly convex, then it is established trivially that every convex set is Chebyshev; hence, on the basis of what was said earlier, as a consequence of Theorem 1 we obtain Theorem 2.

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

The main points of the proof of Theorem 1 are set forth in 7°. In the following paragraph some propositions are given on enlargements of the given set (M) (they are needed for 7°).

6°. Let (X) be an arbitrary Banach space, (M \subset X). The set of points (x \in X) for which (\rho(x,M) < a) ((a>0)) will be called the (a)-extension of the set (M) and will be denoted by the symbol (M_a).

Then:

1) (M_a) is an open set.

2) If (M) is connected, then (M_a) is also connected.

3) If (M) is bounded, then (M_a) is also bounded; the boundary of (M_a) will be denoted by (G_a), and the set (M_a+G_a) by (\overline{M}_a).

4) (G_a) is the set of all (x \in X) for which (\rho(x,M)=a).

5) If (x) does not belong to (M_a), then
[
\rho(x,M_a)=\rho(x,\overline{M}_a)=\rho(x,G_a).
]

6) If (x) does not belong to (M_a), then
[
\rho(x,M)=\rho(x,M_a)+a.
]

7) (M_{a_1+a_2}=(M_{a_1})_{a_2}).

8) If (M) is bounded, then there exists a number (a_0) such that (G_a), for every (a>a_0), has the property of strong star-shapedness, i.e. every ray issuing from some point (p \in X) intersects (G_a) exactly once.

9) If (M) is a set of existence, then (\overline{M}_a) is also a set of existence.

7°. Let (M \subset X) be a bounded Chebyshev set. Consider an arbitrary element (x \in X) not belonging to (M). By the definition of a Chebyshev set, there exists, and moreover a unique, element (y \in M) such that (\rho(x,y)=\rho(x,M)); we shall call this element (y) the projection of (x) onto (M). It is easy to show that every element (x') belonging to the segment ([x,y]) has as its projection onto (M) the same element (y). It is not clear in advance whether the remaining elements of the ray that goes from (y) to (x) will have a common projection onto (M). Clarification of this question is essentially the main goal of the subsequent arguments.

Take some number (a>0) and a point (x \in X) lying outside (\overline{M}_a). We shall call the projection of the point (x) onto (G_a) the intersection of (G_a) with the segment ([x,y]), where (y) is the projection of (x) onto (M). According to what has been said in this paragraph, taking account of 4) from 6°, we conclude that the projection of (x) onto (G_a) is determined uniquely; moreover, if (b>a), then different points of (G_b) cannot be projected to one and the same point of (G_a). Finally, if (M) is compact, then the projection of (x \in G_b) onto (G_a) depends continuously on (x), and also (x) depends continuously on its projection. Thus, under the indicated conditions, the projection of (G_b) onto (G_a) is homeomorphic to (G_b). Suppose that (X) is an (n)-dimensional Banach space-

space. Then, according to 8) and 6°, if (a) is sufficiently large, then (G_a) and all (G_b) ((b>a)) are ((n-1))-dimensional topological spheres; therefore the projection of (G_b) onto (G_a) covers all of (G_a) (for any (b>a)). But in this case, as is easy to show, if (x) does not belong to (\overline{M}_a) and is projected to the point (y\in M), then the whole ray going from (y) to (x) is projected to the point (y). A set (M_a) with this property shall be called a “sun,” and the rays along which the points (x\in \overline{M}_a) are projected onto (M), its rays. Extend the rays of the “sun” inward into (G_a) by an amount (a-\alpha) ((0<\alpha