Uniform estimate of a compact convex set by a ball in an arbitrary norm (Q2710733)

\(R^p\) 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. NEWLINENEWLINENEWLINEThe 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. NEWLINENEWLINENEWLINEThe 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. NEWLINENEWLINENEWLINEThe proofs use convex analysis. Questions of unicity of the minimum, and whether \(x_0\) is in \(D\) are also investigated. NEWLINENEWLINENEWLINESimilar results for the case of Euclidean norm were obtained earlier by the reviewer \textit{I. Bárány} [Acta Sci. Math. 52, No. 1/2, 93-100 (1988; Zbl 0652.52005)], and by \textit{M. S. Nikolskii} and \textit{D. B. Silin} [Tr. Mat. Inst. Steklova 211, 338-354 (1995)].