The authors continue their investigations on best approximation problems in asymmetric normed spaces. An asymmetric norm on a real vector space \(X\) is a functional \(\|\cdot|:X\to[0,\infty)\) satisfying all the axioms of a norm with positive homogeneity instead of absolute homogeneity (see [S. Cobzaş, Functional analysis in asymmetric normed spaces. Basel: Birkhäuser (2013; Zbl 1266.46001)]). For a nonempty subset \(M\) of \(X\) and \(x\in X\), put \(\rho(x,M)=\inf\{\|y-x|: y\in M\}\) (the distance from \(x\) to \(M\)) and \(P_Mx= \{y\in M:\|y-x|=\rho(x,M)\}\) (the metric projection). Due to the asymmetry of the norm \(\|\cdot|\) (that is, the possibility that \(\|{-}x|\ne\|x| \) for some \(x\in X\)), many results from the symmetric case cease to hold in the asymmetric case (e.g., the continuity of the distance function).
The focus in the present paper is on various kinds of solar properties of sets which reveal the structure of approximating sets and, in some cases, can also be useful for designing numerical approximation algorithms.
As the authors point out in the abstract: “We establish a number of theorems of geometric approximation theory in asymmetrically normed spaces. Sets with continuous selection of the near-best approximation operator are studied and properties of such sets are discussed in terms of \(\delta\)-solar points and the distance function. A result on the coincidence of the classes of \(\delta\)- and \(\gamma\)-suns in asymmetric spaces is given. An asymmetric analogue of the Kolmogorov criterion for an element of best approximation for suns, strict suns, and \(\alpha\)-suns is put forward.”