Uniform estimate of a compact convex set by a ball in an arbitrary norm.

*(English. Russian original)*Zbl 0981.52005
Sb. Math. 191, No. 10, 1433-1458 (2000); translation from Mat. Sb. 191, No. 10, 13-38 (2000).

\(R^p\) is equipped with a norm whose unit ball is \(B\). Set \(B(x,r)=x+rB\) and let \(D \subset R^p\) be a compact convex set.

The question of “best uniform approximation” addressed here is to find \(x \in R^p\) and \(r>0\) such that the Hausdorff distance of \(D\) and \(B(x,r)\) is minimal. Define \(R(x)= \max \{ \|x-y\|: y \in D\}\) and \(\rho_A(x)=\min \{\|x-y\|: y \in A\}\). Let \(P(x)=\rho _D(x)-\rho _{R^p\setminus D}(x)\). Both \(R\) and \(P\) turn out to be convex functions.

The main result is that the above approximation problem is equivalent to the following minimization problem: minimize \(R(x)+P(x)\) over all \(x \in R^p\). They are equivalent in the sense that if \((x_0,r_0)\) is a solution to the approximation question, then \(x_0\) is a solution to the minimization question and \(2r_0=R(x_0)-P(x_0)\), and conversely, if \(x_0\) solves the minimization problem, then \((x_0,(R(x_0)-P(x_0))/2)\) solves the approximation problem.

The proofs use convex analysis. Questions of unicity of the minimum, and whether \(x_0\) is in \(D\) are also investigated.

Similar results for the case of Euclidean norm were obtained earlier by the reviewer I. Bárány [Acta Sci. Math. 52, No. 1/2, 93-100 (1988; Zbl 0652.52005)], and by M. S. Nikolskii and D. B. Silin [Tr. Mat. Inst. Steklova 211, 338-354 (1995)].

The question of “best uniform approximation” addressed here is to find \(x \in R^p\) and \(r>0\) such that the Hausdorff distance of \(D\) and \(B(x,r)\) is minimal. Define \(R(x)= \max \{ \|x-y\|: y \in D\}\) and \(\rho_A(x)=\min \{\|x-y\|: y \in A\}\). Let \(P(x)=\rho _D(x)-\rho _{R^p\setminus D}(x)\). Both \(R\) and \(P\) turn out to be convex functions.

The main result is that the above approximation problem is equivalent to the following minimization problem: minimize \(R(x)+P(x)\) over all \(x \in R^p\). They are equivalent in the sense that if \((x_0,r_0)\) is a solution to the approximation question, then \(x_0\) is a solution to the minimization question and \(2r_0=R(x_0)-P(x_0)\), and conversely, if \(x_0\) solves the minimization problem, then \((x_0,(R(x_0)-P(x_0))/2)\) solves the approximation problem.

The proofs use convex analysis. Questions of unicity of the minimum, and whether \(x_0\) is in \(D\) are also investigated.

Similar results for the case of Euclidean norm were obtained earlier by the reviewer I. Bárány [Acta Sci. Math. 52, No. 1/2, 93-100 (1988; Zbl 0652.52005)], and by M. S. Nikolskii and D. B. Silin [Tr. Mat. Inst. Steklova 211, 338-354 (1995)].

Reviewer: I.Bárány (Budapest)