Geometry and topology of continuous best and near best approximations (Q1583970)

Let \(M\) be a given subset of the normed space \(X\) and for any \(x\in X\) let \(P_M(x)\) be the (possibly empty) set of all its best approximations. An \(\varepsilon\)-near best approximation of \(X\) by \(M\) is a function \(\phi:X\rightarrow M\) such that \(\|x-\phi(x)\|\leq d(x,M)+\varepsilon.\) The authors show that if \(X\) is strictly convex, \(M\) is closed, boundedly compact and for each \(\varepsilon > 0\) there exists a continuous \(\varepsilon\)-near approximation \(\phi:X\rightarrow M,\) then \(M\) is a Chebyshev set. Sufficient conditions implying the connectedness of \(P_M(x)\) are given. They obtain upper bounds, (depending on the modulus of convexity of \(X\)), for the Chebyshev radius of \(P_M(x)\) with respect to a convenient point \(x^\prime(x).\)