L. P. VLASOV

ON CHEBYSHEV SETS IN BANACH SPACES

(Presented by Academician P. S. Novikov, 3 VI 1961)

We use the terminology introduced by N. V. Efimov and S. B. Stechkin \((^{1,2})\).

A set \(M\) of a Banach space \(X\) is called Chebyshev if for every point \(x \in X\) there is in \(M\) a unique point \(y\) nearest to \(x\), i.e., such that \(\|x-y\|=\rho(x,M)\). The point \(y\) is then called the projection of the element \(x\) onto the set \(M\). A Chebyshev set \(M\) is called a sun if, for every point \(x \in X\), the entire ray \(yx\) (issuing from \(y\) and passing through \(x\)) is projected into the point \(y\)—the projection of \(x\) onto \(M\). A set is called boundedly compact if its intersection with every closed ball is compact in itself.

Under certain conditions the problem of Chebyshev sets admits an extremely simple solution, if one uses the fixed-point principle. (For an application of this principle to Chebyshev sets, see also the paper of V. Klee \((^3)\).)

Theorem. In an arbitrary Banach space every boundedly compact Chebyshev set is a sun.

An analogous proposition was proved by N. V. Efimov and S. B. Stechkin in the case of a uniformly convex space.

Proof. Suppose the contrary, i.e., that the set \(M\) is not a sun. Then there exists a point \(x\) such that the ray \(yx\) (\(y\) is the projection of \(x\) onto \(M\)) is not entirely projected into the point \(y\). Obviously, on the ray \(yx\) there is a point farthest from \(y\) that is still projected into \(y\) (for convenience denote it again by \(x\)). Consider some closed ball \(V\) with center at the point \(x\), not intersecting the set \(M\). Denote its bounding sphere by \(S\).

Introduce a mapping \(\varphi\) (\(z \to z''\)) of the ball \(V\) into itself as follows. For a point \(z \in V\) find its projection \(z'\); then, of the two points of intersection of the ray \(z'x\) with the sphere \(S\), take the point \(z''\) lying outside the segment \([z',x]\), and put \(\varphi(z)=z''\). It is clear that \(\varphi\) is a continuous mapping (otherwise, for a point of discontinuity, by virtue of the bounded compactness of the set \(M\), there would exist two nearest points in \(M\)) and that the set \(\varphi(V)\) is compact (since \(M\) is boundedly compact).

By Schauder’s principle, in a Banach space every continuous mapping of a convex closed set into its compact part has a fixed point (see, for example, \((^4)\), p. 578). In our case the closed ball \(V\) is mapped into its compact part, \(\varphi(V)\). Therefore there exists a point \(z_0\) such that \(z_0=\varphi(z_0)\). If \(z'_0\) is the projection of \(z_0\) onto \(M\), then this equality means that the segment \([z_0,z'_0]\) contains the point \(x\) and \(z_0 \in S\). But then the point \(z'_0\) will also be nearest to \(x\), and since \(y\) is also nearest to \(x\), it follows that \(z'_0=y\). The point \(z_0\) lies on the ray \(yx\) outside the segment \([y,x]\) and is projected into \(y\). This is impossible by the assumption that \(x\) is the last point of the ray \(yx\) projected into \(y\). Thus the theorem is proved.

Lemma. In a smooth Banach space every sun is a convex set.

This fact was noted by N. V. Efimov and S. B. Stechkin. For completeness we give the proof.

Suppose that the sun \(M\) is not convex. Then there exist points \(a, b, c\) such that

\[ a \in M,\qquad b \in M,\qquad c \notin M,\qquad c \in [a,b]. \]

Let the point nearest to \(c\) in \(M\) be the point \(p\). At least one of the segments \([a,p]\), \([b,p]\) intersects the interior of the ball \(V\) with center at \(c\) and radius \(\|c-p\|\). Otherwise, on the basis of the Hahn–Banach theorem, through each of the segments \([a,p]\), \([b,p]\) one could draw a supporting hyperplane to the ball \(V\). At the point \(p\) of the ball we would then have two supporting hyperplanes, and this would mean nonsmoothness of the space, contrary to the assumption. Thus, let, for example, the segment \([b,p]\) intersect the interior of the ball \(V\). Then for some ball \(V'\), similar to the ball \(V\) with center of similarity at the point \(p\), the point \(b\) is an interior point. But the center \(c'\) of the ball \(V'\) lies on the ray \(pc\), and the point \(b\) is closer to \(c'\) than \(p\) is. This contradicts the fact that \(M\) is a sun. The lemma is proved.

From the facts proved there follows the following

Theorem. In every smooth Banach space every Chebyshev boundedly compact set is convex.

Special cases of this theorem were proved in \((^{2,3})\).

Ural State University
named after A. M. Gorky

Received
25 V 1961

REFERENCES

\(^{1}\) N. V. Efimov, S. B. Stechkin, DAN, 118, No. 1, 17 (1958).
\(^{2}\) N. V. Efimov, S. B. Stechkin, DAN, 127, No. 2, 254 (1959).
\(^{3}\) V. Klee, Math. Ann., 142, 292 (1961).
\(^{4}\) L. V. Kantorovich, G. P. Akilov, Functional Analysis in Normed Spaces, Moscow, 1959.