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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)].

##### MSC:
 52A40 Inequalities and extremum problems involving convexity in convex geometry 52A27 Approximation by convex sets 90C25 Convex programming
##### Keywords:
convex sets; best uniform approximation; convex analysis
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