MATHEMATICS

P. K. BELOBROV

ON THE CHEBYSHEV POINT OF A SYSTEM OF PLANES IN A BANACH SPACE

(Presented by Academician L. V. Kantorovich, 18 IX 1969)

We consider the problem of the minimum of a convex, continuous, and nonnegative functional (f(x)), defined on a Banach space (X), for which the set
[
L={x\mid f(x)=f(0),\ x\in X}
]
is linear, and moreover
[
f(z+x)=f(z)
]
for every element (x\in L). We shall call the set (L) the subspace of constancy of the functional (f(x)). An example of a functional of this type is
[
\sup_{i\in I}\rho(x,H_i),
]
where (H_i=x_i+L_i,\ i\in I), (L_i) is a subspace of (X), (x_i\in X), and (\rho(x,H_i)) is the distance from (x) to (H_i). For this functional the subspace of constancy is
[
\bigcap_{i\in I} L_i,
]
and a point (x^\in X) for which
[
\sup_{i\in I}\rho(x^,H_i)=\inf_{x\in X}\sup_{i\in I}\rho(x,H_i)
]
is called a Chebyshev point for the system of planes (H_i,\ i\in I). Algorithms for finding such a point in the case of a finite-dimensional space were developed by S. I. Zukhovitskii ((^1)); a proof of its existence for finite (I={1,\ldots,n}), an arbitrary Banach space, and (L_i,\ i\in I), of unit defect (index), was given by A. L. Garkavi ((^2)).

Another important example of a functional possessing a subspace of constancy is (|Ux-y|), where (U) is a linear and continuous operator from (X) into a Banach space (Y), and (y) is a fixed element of (Y); for this functional the subspace of constancy is the set of zeros of the operator (U).

We state the following condition:

((1)). For every number (r>0) there exists a number (R>0) such that from the inequality (f(x)\le r) it follows that (\rho(x,L)\le R).

For the functional
[
\sup_{i\in I}\rho(x,H_i)
]
this condition is formulated as follows:

((1+)). For every number (r>0) there exists a number (R>0) such that from the inequalities
[
\rho(x,L_i)\le r,\quad i\in I,
]
it follows that
[
\rho!\left(x,\bigcap_{i\in I}L_i\right)\le R.
]

For condition (1) to hold for the functional (|Ux-y|), it is sufficient, and if the subspace of (U) is quotient-reflexive,* also necessary, that the range of the operator (U) be closed.

Theorem 1. In order that condition ((1+)) hold for the subspaces (L_1,\ldots,L_m), it is sufficient (and for (m=2) also necessary) that the sum of the subspaces
[
\sum_{i=1}^{m}L_i
]
be closed.

For (m>2) condition ((1+)) is no longer sufficient for the closedness of the sum of the subspaces.

Theorem 2. If for a convex, continuous, and nonnegative functional (f(x)) the subspace of constancy is quotient-reflexive and

* A subspace (L\subset X) is called quotient-reflexive ((^3)) if the quotient space (X/L) is reflexive.

satisfies condition (1), then there exists a point (x^ \in X) such that
[
\inf_{x\in X} f(x)=f(x^).
]

From Theorems 1 and 2 there follows

Corollary. Let (\bigcap_{i=1}^m L_i) be a factor-reflexive subspace. In order that, for any planes (H_i=x_i+L_i,\ x_i\in X,\ i=1,\ldots,m), there exist a Chebyshev point, it is sufficient (and for (m=2) also necessary) that the sum (\sum_{i=1}^m L_i) be closed (that condition ((1+)) be satisfied).

For (m>2) these conditions are not necessary.

Let, further, (A_i) be a linear and continuous operator from (X) into the Banach space (Y_i); (D_i, N_i, R_i) respectively the domain of definition, the null set, and the range of the operator (A_i,\ i=1,\ldots,m,\ \bigcap_{i=1}^m D_i\ne \Lambda).

Denote by (\varphi(\ldots)) some norm in (m)-dimensional coordinate space.

Theorem 3. If the space (\bigcap_{i=1}^m N_i) is factor-reflexive and the sets (\sum_{i=1}^m N_i,\ \bigcap_{i=1}^m D_i,\ R_i,\ i=1,\ldots,m,) are closed, then in (\bigcap_{i=1}^m D_i) there exists a point delivering the minimum to the functional
[
\varphi(|A_1x-y_1|,\ldots,|A_mx-y_m|),
]
whatever fixed elements (y_i\in Y_i,\ i=1,\ldots,m), may be.

Theorem 4. Whatever the system of functionals (f_1,\ldots,f_n \subset X^) and the system of numbers (c_1,\ldots,c_n,d), where (d\ge 0), there exists a system of points (x_1^,\ldots,x_m^) ((m) fixed), for which
[
\max_{1\le i\le n}\min_{1\le j\le m}|f_i(x_j^)-c_2|
=
\inf_{\substack{x_1,\ldots,x_m\subset X\ \max_{i,j\le m}|x_i-x_j|\le d}}
\max_{1\le i\le n}\min_{1\le j\le m}|f_i(x_j)-c_i|.
]

Hence, for (m=1) and (d=0) (as, however, also from the corollary and Theorem 3), there follows the existence of a Chebyshev point of a finite system of hyperplanes. The problem of finding such a point in the present case can be reduced to solving a certain system of equations. Namely, let the system of functionals (f_1,\ldots,f_n \subset X^*) be linearly independent,
[
f_{n+i}=\sum_{k=1}^n \lambda_k^{(i)} f_k,
]
(i=1,\ldots,m), and let the system of numbers (c_1,\ldots,c_{n+m}) be arbitrary. Denote
[
\mu_i=c_{n+i}-\sum_{k=1}^n \lambda_k^{(i)}c_k,\qquad i=1,\ldots,m,
]
and suppose, for example, that (\mu_1\ne 0). By the method of linear programming we find the quantity
[
N=\max\left(\sum_{k=1}^n \lambda_k^{(1)}x_k-x_{n+1}\right),
]
where the maximum is taken over all systems of numbers (x_1,\ldots,x_{n+m}) satisfying the equation
[
\sum_{k=1}^n\left(\lambda_k^{(i)}-\frac{\mu_i}{\mu_1}\lambda_k^{(1)}\right)x_k
+\frac{\mu_i}{\mu_1}x_{n+1}-x_{n+i}=0,\qquad i=2,\ldots,m,
]
and the inequalities (|x_k|\le 1,\ k=1,\ldots,n+m); moreover, we find a system of numbers (x_1^0,\ldots,x_{n+m}^0), at which the quantity (N) is attained.

Theorem 5. Every solution of the determined system

[
f_\nu(x)=c_\nu-\left(\sum_{k=1}^{n}\lambda_k^{(1)}c_k-c_{n+1}\right)\frac{x_\nu^0}{N},\qquad \nu=1,\ldots,n,
]

is a Chebyshev point for the system of hyperplanes (f_j(x)=c_j), (j=1,\ldots,n,\ n+1,\ldots,n+m). If the system of numbers (x_1^0,\ldots,x_{n+m}^0) is unique, then the converse is also true.

I express my gratitude to A. L. Garkavi for his attention to the work and valuable advice.

Rostov Civil Engineering Institute

Received
16 VI 1969

REFERENCES

1. S. I. Zukhovitskii, DAN, 139, No. 3 (1961).
2. A. L. Garkavi, in: Collected Mathematical Papers, V. V. Kuibyshev Military Engineering Academy, 1968.
3. A. L. Garkavi, Mat. sbornik, 62 (104), No. 1 (1963).